LIBRARY 

UNIVERSITY  Of 
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UNIVERSITY   OF   CAUFORNlf     Y||  iri'lllli  I  11 


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COLLEGE   ALGEBRA 


WITH 


APPLICATIONS 


BY 

E.    J.    WILCZYNSKI,  Ph.D. 

THE    UNIVERSITY    OF    CHICAGO 


EDITED    BY 

H.    E.    SLAUGHT,  Ph.D. 

THE    UNIVERSITY    OF    CHICAGO 


ALLYN    AND    BACON 
23oston  NebJ  gork  Ctirago 


COPYRIGHT,   1916,  BY 
E.  J.  WILCZYNSKl 


Noriuooti  Prtss 

J.  S.  Gushing  Co.  — Berwick  &  Smith  Co. 

Norwood,  Mass.,  U.S.A. 


PREFACE 

Nothing  causes  more  trouble  to  young  inatliematicians  than  the 
traditional  method  of  introducing  complex  numbers  into  algebra. 
The  student  knows  that  the  square  root  of  minus  one  is  neither 
a  positive  nor  a  negative  number,  and  that  it  is  not  equal  to  zero. 
Nobody  has  ever  told  him  that  there  are  any  numbers  which  are 
neither  positive,  nor  negative,  nor  zero.  In  spite  of  this,  square 
roots  of  negative  numbers  are  introduced  for  him  to  reckon  with. 
Of  course  he  does  not  know  what  they  mean  and  he  becomes  sus- 
picious. Even  later,  if  a  concrete  representation  of  these  so- 
called  "  imaginary  "  numbers  is  ever  presented  to  him,  he  never 
quite  gets  over  his  first  suspicions  ;  the  uncomfortable  feeling  re- 
mains within  him  that,  somehow,  he  has  been  cheated,  that  imag- 
inary numbers  are  impossible  and  without  meaning. 

It  is  historically  correct  to  introduce  imaginaries  in  this  way. 
But  the  student's  doubts  are  also  justified  historically.  His  sus- 
picious attitude  toward  imaginary  numbers  is  merely  a  repeti- 
tion of  the  position  taken  by  the  whole  mathematical  world  from 
the  fifteenth  to  the  nineteenth  century. 

The  historical  order  of  presentation  is  not  ahvays  the  best  peda- 
gogic order,  and  the  subject  of  complex  numbers  is  one  of  the  most 
striking  instances  ichere  the  historical  order  should  be  avoided. 

In  Chapter  I  of  the  present  book,  the  imniber  system  of  algebra, 
up  to  and  including  the  system  of  complex  numbers,  is  developed 
in  a  concrete  and  convincing  way  by  means  of  the  geometry  of 
directed  line-segments  or  vectors.  No  student  will  feel  any  doubt 
concerning  the  legitimacy  of  complex  numbers,  after  mastering 
this  chapter,  unless  his  suspicions  have  been  previously  aroused  in 
his  first  course  in  algebra.  In  a  sense,  Chapter  I  may  be  regarded 
as  devoted  to  the  foundations  of  algebra.  It  is  not,  however,  a 
chapter  on  foundations  in  the  formal  technical  sense  in  which 
that  word  is  being  used  at  the  present  time. 

Chapter  I  also  serves  the  purpose  of  revieicing  the  most  ele- 


iv  PREFACE 

mentary  parts  of  arithmetic  and  algebra.  A  revieio,  tvJiich  is  a 
mere  repetition  of  material  covered  once  before,  is  a  great  ivaste  of 
time.  For  this  reason  no  formal  review  work  is  offered  at  the  be- 
ginning of  the  book.  The  review  work  is  scattered  through  the 
various  chapters.  It  is  placed  wherever  it  is  needed  for  the  pur- 
poses in  hand,  and  it  is  always  illuminated  by  the  discussions 
which  precede  and  follow  it.  That  everything  really  essential  is 
covered  by  this  kind  of  a  review  is  guaranteed  by  the  nature  of 
the  book  which,  while  intended  for  use  in  college,  is  built  up  from 
iirst  principles,  so  that  no  reference  to  more  elementary  books  on 
algebra  is  ever  necessary.  A  fairly  mature  mind  might  begin  his 
study  of  algebra  with  this  book. 

The  principal  criticism  of  "  college  algebra "  as  a  course  has 
been  its  lack  of  unity.  This  criticism  seems  to  be  exceedingly 
well  founded,  if  we  examine  the  various  texts  on  the  subject 
which  have  appeared  to  date.  The  present  book  ivas  vritten  pri- 
marily for  the  purpose  ofshoiving,  that  the  undoubted  scrajjpiness  of  the 
traditional  course  in  college  algebra  is  due  to  poor  arrangement  and 
piresentation,  and  not  to  any  intrinsic  defect  of  the  subject  itself.  The 
following  sketch  of  the  contents  of  the  book  will  show  how  the 
desired  unification  has  been  accomplished. 

The  function  concept  is  the  central  notion  of  the  book.  Chapter  I, 
as  has  been  mentioned,  is  devoted  to  the  number  system,  a  neces- 
sary preliminary.  Chapter  II  begins  with  a  general  discussion  of 
the  function  concept  and  takes  up  in  detail  the  simplest  cases, 
namely  the  subject  of  variation,  and  linear  functions  of  a  single 
variable,  the  graphs  of  such  functions  being  introduced  at  an 
early  stage.  Chapter  III  deals  with  quadratic  functions  and 
equations.  Chapter  IV  with  integral  rational  functions  of  any 
order,  the  corresponding  equations,  and  the  numerical  calculation 
of  their  roots  (theory  of  equations)  ;  this  treatment  is  completed 
in  (Jhapter  V  by  a  discussion  of  the  algebraic  calculation  of  their 
roots,  the  fundamental  theorem,  and  so  on.  Chapter  VI  deals 
with  fractional  rational  functions,  in  particular  with  their  expres- 
sion as  a  sum  of  partial  fractions.  Chapter  VII  discusses  the 
simplest  irrational  functions  and  formulates  the  distinction  be- 
tween algebraic  and  transcendental  functions.  Chapter  VIII 
follows  with  the  general  power  function,  the  exponential  func- 
tion, and  logarithms. 

In  Chapter  IX  functions  of  more  than  one  variable  are  intro- 


PREFACE  V 

(lueed,  linear  functions  to  begin  with,  leading  to  determinants  of 
the  second  and  third  order.  In  preparation  for  the  general 
theory  of  determinants  of  the  ?ith  order,  we  interpolate  a 
chapter  (Chapter  X)  on  permutations  and  combinations,  followed 
by  Chapter  XI  on  probability.  We  are  now  ready  for  Chapter 
XII,  which  discusses  linear  functions  of  oi  variables  and  determi- 
nants of  the  nth  order.  Chapter  XIII  goes  on  with  the  discussion 
of  quadratic  functions  of  two  independent  variables,  and  simul- 
taneous quadratics. 

So  far  the  functions  considered  have  been  functions  of  contin- 
uous variables.  If,  instead,  we  restrict  the  variable  to  integral 
values,  every  function  gives  rise  to  a  sequence.  Thus  a  linear 
function  gives  rise  to  an  arithmetic  progression  and,  for  this 
reason,  arithmetic  progressions  are  included  in  Chapter  II.  Har- 
monic and  geometric  progressions,  suggested  by  analogy,  are  also 
treated  in  Chapter  II. 

In  Chapter  XIV,  however,  the  discontinuous  variable  is  em- 
phasized, and  results  in  a  more  extensive  discussion  of  sequences 
and  series  with  a  finite  number  of  terms.  Chapter  XV  and  XVI, 
on  limits  and  series,  now  follow  naturally  from  the  suggestions 
of  Chapter  XIV  and  earlier  chapters. 

Tlie  applications  are  scattered  through  the  entire  hook  and  form  an 
integral  part  of  it.  They  are  discussed  with  as  much  care  as 
though  the  book  had  been  written  for  their  sake.  Only  ajjpli- 
cations  of  real  and  general  importance  have  been  included,  making 
the  course  in  algebra  a  valuable  adjunct  to  the  courses  in  physics  and 
chemistry.  These  applications  include  such  subjects  as  the  meas- 
urement of  length,  time,  and  mass  ;  the  theory  of  the  vernier,  slide 
rule,  logarithmic  paper,  and  of  scales  in  general ;  the  notions  of 
velocity,  acceleration,  density,  specific  gravity,  force,  uniform 
motion,  uniformly  accelerated  motion,  pressure  of  gases ;  the 
principle  of  Archimedes,  the  motion  of  a  projectile,  Doppler's 
principle,  the  theory  of  dimensions  in  physics,  and  indirect  analy- 
sis in  chemistry.  They  also  include  a  discussion  of  compound 
interest,  annuities,  and  life  insurance  ;  the  comj)Oun(l  interest  law 
and  its  applications  to  dampened  vibrations,  transmission  of 
light,  pressure  in  the  atmosphere,  and  cooling  bodies.  Professor 
A.  C.  Lunn  was  kind  enough  to  give  the  author  the  benefit  of  his 
criticisms  on  some  of  these  topics,  and  we  take  this  opportunity  of 
expressing  our  indebtedness  to  him. 


vi  PREFACE 

Each  of  these  applications  is  discussed  as  carefully  as  though  the 
book  ivere  a  treatise  on  chemistry  or  physics.  The  student  is  never 
asked  to  do  examples  involving  applications  which  he  cannot 
understand  because  the  fundamental  principles  are  not  explained. 
Of  course,  these  applications  are  here  classified  from  the  mathe- 
matical point  of  view,  and  therefore  they  appear  in  an  order 
different  from  that  which  would  result  if  the  point  of  view  were 
primarily  physical  or  chemical.  But  the  student  can  only  gain 
by  having  the  same  subject  appear  in  such  a  different  way  in 
several  of  his  courses. 

AVe  have  explained  the  general  policy  of  the  book.  Much  more 
might  be  said  about  specific  details  in  which  it  differs  from  other 
books  ;  we  hope  that  the  reader  will  discover  these  for  himself, 
and  thus  make  it  unnecessar}'  for  us  to  unduly  expand  this 
preface.  We  close  with  the  request  that  all  prospective  users  of 
the  book  read  the  suggestions  to  the  instructor  and  to  the  student 
given  on  pages  vii  and  ix.  An  answer  book  will  be  supplied  to 
those  classes  whose  instructors  wish  this  to  be  done. 

E.    J.    WILCZYNSKI. 

H.    E.    SLAUGHT,  Editor. 


SUGGESTIONS   TO   THE   INSTRUCTOR 

The  material  incliuled  in  this  book  probably  contains  everything  ever 
given  under  the  title  "  College  Algebra"  in  any  American  college.  But  it 
includes  more  than  can  be  given  profitably  in  any  one  course  of  about 
fifty  recitations.  It  will  usually  be  necessary  to  make  a  selection.  The 
following  sample  outlines  of  courses  are  intended  to  be  helpful  in  this 
connection.  But  it  should  be  understood  that,  although  they  have  been 
considered  carefully,  many  different  selections  might  be  made  by  the 
instructor,  who  probably  knows  the  special  needs  of  his  class  better  than 
the  author. 

Another  possible  way  of  using  the  book  would  be  to  consolidate  the 
courses  in  College  Algebra  and  Analytic  Geometry,  supplementing  the  book 
with  explanations  by  the  teacher  on  such  topics  in  analytic  geometry  as 
are  not  included.  It  may  also  be  used  as  a  basis  for  a  course  in  higher 
algebra. 

Course  A 

Short  course.     Emphasis  upon  the  applications. 
Articles  16-75,  78-113,   126-132,  135-146,  148-151,  156-182, 
185-212. 

Much  of  this  work  has  been  covered  in  the  student's  high  school  course 
and  may  be  reviewed  in  a  brief  time.  Arts.  167-17.5  (on  calculation  with 
logarithms)  may  be  transferred  to  the  course  in  trigonometry.  Arts.  80, 
195,  190  may  be  omitted. 

The  subjects  included  in  Course  A  may  be  treated  in  the  order  given  in 
the  book,  or  else  in  the  order :  Chapters  I,  II,  IX,  III,  IV,  etc. ;  or  Chap- 
ters I,  II,  III,  IX,  IV,  etc. 

Course  B 

To  Course  A  add  articles  213-217,  219-227,  238-240,  243-252, 
256,  on  permutations,  combinations,  probability,  and  simultaneous 
quadratics. 

Course  C 

To  Course  A  add  articles  257-297,  on  summation  of  series, 
limits,  and  infinite  series. 


viii  SUGGESTIONS   TO   THE  INSTRUCTOR 

Course  D 

Short  course.     Emphasis  upon  the  purely  mathematical  aspects. 
Articles    1-38,    50-78,    82-111,    114-120,    126-145,    147-184, 
197-210. 

See  remarks  under  Course  A. 

Course  E 

To  Course  D  add  articles  213-217,  219-227,  238-240,  243-252, 
256,  on  permutations,  combinations,  probability,  and  simultaneous 
quadratics. 

Course  F 

To  Course  D  add  articles  257-297,  on  summation  of  series, 
limits,  and  infinite  series. 


SUGGESTIONS   TO   THE   STUDENT 

Material 

The  student  should  provide  himself  with  a  pair  of  compasses,  a 
ruler  divided  decimally  into  centimeters  and  millimeters,  and  a 
quantity  of  cross-section  paper  (preferably  millimeter  paper),  for 
facilitating  his  graphic  work. 

General  Directions  for  Study 

1.  Carefully  read  the  assigned  lesson. 

2.  Test  your  understanding  of  the  j)rinciples  involved  as 
follows  : 

(a)  Convince  yourself  that  you  are  really  able  to  follow  the 

logic  of  the  argument  by  stating,  explicitly,  a  reason 
in  justification  of  every  one  of  its  steps. 

(b)  Try  to  discover  a  fundamental  idea  which  runs  through 

the  argument  and  illuminates  it.  This,  together  with 
(a),  will  enable  you  to  master  the  whole  argument,  so 
as  to  reproduce  it  and  to  use  similar  reasoning,  else- 
where, under  similar  circumstances. 

(c)  See  whether  you  can  appreciate  the  practical  importance 

of  the  subject  under  discussion  by  finding  some  appli- 
cations of  it. 

3.  If  this  test  of  yourself,  recommended  in  No.  2,  turns  out  to 
your  own  satisfaction,  proceed  to  the  examples. 

4.  If  your  self-examination -has  had  an  unsatisfactory  result, 
find  out  whether  your  difficulty  should  be  classified  under  2a,  2b, 
or  2c.  If  it  falls  under  2a,  try  to  find  out  just  where  the  argument 
begins  to  become  unintelligible  for  the  first  time,  and  fix  your  at- 
tention on  the  statements  made  at  that  ])oint.  Look  up,  in 
review,  the  definitions  of  the  terms  which  are  used  in  those  state- 
ments. Nothing  causes  more  trouble  in  mathematics  than  lack 
of  appreciation  of  just  what  the  definitions  say.     If   you  find 


X  SUGGESTIONS  TO   THE   STUDENT 

that  your  trouble  is  not  with  the  definitions,  search  your  memory 
and  your  text  for  passages  dealing  with  the  subjects  that  cause 
the  difficulty.  Use  the  index  and  the  table  of  contents.  Finally, 
if  you  fail  to  conquer  the  difficulty,  formulate  it  in  writing  as  a 
question  to  your  instructor.  Often  you  will  be  able  to  answer 
your  question  yourself  after  having  put  it  into  written  form. 

If  your  difficulty  is  not  with  2a,  but  with  26,  or  2c,  tr}^  the  ex- 
amples.    They  will  probably  helj)  you. 

5.  Make  your  oral  and  written  statements  clear  and  unambig- 
uous. As  a  test  of  clearness,  imagine  yourself  addressing  a 
person  of  intelligence  who  is  properly  prepared  to  follow  your 
argument,  but  do  not  expect  him  to  do  any  mind-reading.  Ask 
yourself  this  question :  Can  such  a  person  be  expected  to  under- 
stand what  I  have  said  ?  In  written  work  give  everything  that 
is  essential,  and  try  to  make  no  statement  which  is  not  literally 
true.  Good  paper,  good  ink,  and  neatness  in  form  are  of  great 
assistance  in  orderly  thinking, 

6.  Formulate  in  writing  questions  concerning  the  lesson.  Pre- 
sent them  to  your  instructor.  But  try  first  to  answer  them  for 
yourself. 

7.  Some  students  rush  to  the  examples  without  reading  the 
text.  You  will  fail  if  you  adopt  this  plan.  Even  if  you  should 
succeed  in  solving  the  examples,  you  will  probably  fail  to  under- 
stand the  principles  which  they  are  intended  to  illustrate.  It  is 
not  the  purpose  of  a  course  in  algebra  to  train  you  to  merely  fol- 
low mechanically  certain  rules  of  procedure.  Make  the  rules 
your  own  by  practice  and  memory ;  but  more  important,  make 
them  truly  and  permanently  your  property  by  means  of  a  thorough 
understanding  of  the  principles  upon  which  they  are  based. 


CONTENTS 


CHAPTER   I 


The  Number  System  of  Algebra. 


7. 

8. 

9. 
10. 
11, 
12. 

13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 
22. 
23. 
24. 
25. 
26. 
27. 
28. 

29. 


The  positive  integer 

Addition,  subtraction,  and  multiplication  of  positive  integers 
Factors  or  divisors  of  integers    . 

Division  of  integers 

The  greatest  common  divisor  of  two  integers 

The  prime  factors  of  an  integer  . 

Geometric  representation  of  positive  integers 

Rational  numbers 

Some  properties  of  rational  numbers 

Addition  and  subtraction  of  rational  numbers 

Multiplication  of  rational  numbers     . 

Geometric  construction  for  the  product  of  two  positive  rational 

numbers       ...... 

Division  of  positive  rational  numbers 

A  further  property  of  rational  numbers     . 

The  existence  of  irrational  numbers  . 

Negative  numbers  and  zero 

Directed  lines  and  directed  line-segments  . 

Addition  and  subtraction  of  positive  and  negative  numbers 

Multiplication  of  positive  and  negative  numbers 

Division  by  a  positive  or  negative  divisor  .... 

Division  by  zero 

The  monotonic  laws  for  positive  and  negative  numbers     . 
Directed  line-segments  in  a  plane       ..... 

The  complex  numbers 

Equality  of  two  complex  numbers      ..... 
Vectors,  vector  addition,  and  addition  of  complex  numbers 
Subtraction  of  vectors  and  complex  numbers     . 
Multiplication  of  a  complex  number  by  a  positive  or  negative 

number 

Multiplication  of  a  complex  number  by  i  . 


PAGE 
1 
1 

3 

4 
5 
6 
7 
8 
9 
11 
13 

13 
14 
16 
17 
20 
21 
24 
26 
28 
29 
29 
31 
34 
36 
36 
40 

41 
42 


Xll 


CONTENTS 


30.  Polar  form  of  a  complex  number 

31.  Multiplication  of  two  complex  numbers 

32.  Division  of  complex  numbers 

33.  Real  and  imaginary  numbers 

34.  Conjugate  complex  numbers 

35.  Validity  of  the  fundamental  laws  for  complex  numbers 

36.  History  of  the  number  system  of  Algebra 


CHAPTER  II 


Linear  Functions  and  Progressions. 

37.  Constants  and  variables 

38.  Variation 

39.  Application  to  concrete  problems 

40.  Measurement  of  length 

41.  Length,  area,  and  volume 

42.  Time 

43.  Mass 

44.  Density  and  specific  gravity 

45.  Velocity 

46.  Acceleration        .        . 

47.  Uniformly  accelerated  motion    . 

48.  Falling  bodies      .... 

49.  The  importance  of  dimensional  symbols    . 

50.  Graphical  representation  of  a  pair  of  numbers 

51.  Graphical  representation  of  variation 

52.  Graphical  representation  of  the  function  y  =  nix  +  b 

53.  Slope  of  a  straight  line        .... 

54.  Equation  of  a  straight  line 

55.  The  zero  of  a  linear  function 

56.  Arithmetic  progressions      .... 

57.  Insertion  of  arithmetic  means     . 
68.  Harmonic  progressions       .... 

59.  Geometric  progressions       .... 

60.  Sum  of  a  geometric  progression  of  n  terms 

61.  Geometric  means 

62.  Geometric  progressions  with  infinitely  many  ter 

63.  Periodic  decimals 


CHAPTER   III 

Quadratic  Functions  and  Equations. 

04.    Standard  form  of  a  quadratic  function 
65.   Graph  of  a  quadratic  function    . 


94 
94 


CONTENTS 


Xlll 


66.  Tlie  maximum  or  minimum  of  a  quadratic  function 

67.  Graphic  determination  of  tlie  zeros  of  a  quadratic  function 

68.  Calculation  of  the  real  zeros  of  a  quadratic  function 

69.  Another  method  of  deriving  the  formulpe  for  the  roots  of  a 

quadratic  equation 

70.  Complex  roots  of  a  (juadratic  equation 

71.  Various  methods  of  solving  a  quadratic  equation  or  of  factor 

ing  a  quadratic  function       .... 

72.  Special  forms  of  quadratic  equations 

73.  Equations  of  higher  degree  solvable  by  means  of  (juadratics 

74.  Rational  and  irrational  roots  of  a  quadratic  equation 

75.  Quadratic  surds 

76.  The  square  root  of  an  expression  of  the  form  a  +  h\/d 
11.    Application  of  the  monotonic  laws  of  Algebra  in  numerical 

calculations  involving  quadratic  surds 

78.  Interpretation  of  negative,  fractional,  and  complex  roots  in 

concrete  problems 

79.  Uniform  motion  along  a  straight  line 

80.  Force 

81.  Motion  of  a  projectile  under  the  influence  of  gravity 


PAGB 

96 

99 

100 

102 
104 

106 
109 
110 
111 
112 
115 

117 

118 
120 
121 
126 


CHAPTER   IV 


Integral  Rational  Functions  of  the  nth  Order, 
of  their  Real  Zeros. 


Numerical  Calculation 


82.  Calculation  of  the  numerical  values  of  an  integi-al   rational 

function        ..... 

83.  The  functional  notation 

84.  The  factor  theorem      .... 

85.  The  remainder  theorem 

86.  Synthetic  division       .... 

87.  The  slope  of  the  tangent     . 

88.  The  binomial  theorem 

89.  The  derivative  of  an  integral  rational  function 

90.  Derivatives  of  higher  order  .... 

91.  Taylor's  expansion 

92.  Diminishing  the  roots  of  an  equation 

93.  Arrangement  of  the  calculation 

94.  Multiplication  of  the  roots  of  an  equation  by  m 

95.  Changing  the  sign  of  the  roots    .... 

96.  Continuity  of  integral  rational  functions    . 

97.  Newton's  method  of  approximation    . 

98.  Geometric  significance  of  Newton's  method 

99.  The  method  of  false  position  (Regula  falsi) 


129 
131 
132 
134 
135 
136 
141 
146 
147 
148 
149 
151 
153 
154 
155 
157 
158 
158 


xiv  CONTENTS 


PAGE 


100.  An  example  of  Newton's  method 160 

101.  Horner's  method 164 

102.  Abbreviated  calculation 165 

103.  Negative  roots 165 

104.  Computation  of  more  than  one  root 166 

105.  Upper  limit  for  the  positive  roots  of  an  equation     .        .         .  166 

106.  Descartes's  rule  of  signs 168 

107.  Maxima  and  minima  of  an  integral  rational  function       .         .  172 

108.  Rolle's  theorem .         .         .175 

109.  Multiple  roots 175 

110.  Rational  roots  of  an  equation  with  rational  coefficients  .         .  177 

111.  Summary  of  the  operations  required  in  solving  an  equation 

with  given  numerical  coefficients 181 

112.  Application  of  cubic  equations  to  floating  spheres   .         .         .  182 

113.  Application  of  cubic  equations  in  trigonometry       .        ,        .  185 


CHAPTER   V 

Integral  Rational  Functions  of  the  nth  Order.  The  Problem  of  the 
Algebraic  Determination  of  their  Zeros,  and  their  General 
Properties. 

114.  Distinction  between  the  algebraic  and  numerical  solution  of 

an  equation        .........  187 

115.  The  equation  x"  —  1  =  0 187 

116.  The  nth  power  of  a  complex  number 188 

117.  The  complex  roots  of  unity 189 

118.  Numerical  expressions  for  the  complex  roots   of   unity  for 

n  =  2,  3,  4 191 

119.  Construction  of  regular  polygons 193 

120.  The  equation  x"  —  a  =  0 194 

121.  The  cubic  equation 197 

122.  Discussion  of  the  roots 202 

123.  The  ratios  of  the  coefficients  of  the  general  cubic  equation 

expressed  in  terms  of  its  roots 202 

124.  The  equation  of  the  fourth  order 204 

125.  The  equations  of  higher  order 207 

126.  The  fundamental  theorem  of  Algebra 208 

127.  Application  of  the  fundamental  theorem  to  functions  with 

real  coefficients  .........  212 

128.  Use  of  the  factored  form  of  f(x)  in  plotting    ....  215 

129.  Form  of  the  graph  in  the  case  of  real  and  distinct  factors        .  216 

130.  Form  of  the  graph  in  the  case  of  real  factors  some  of  which 

are  repeated 217 


CONTENTS  XV 


PAGE 


131.  Form  of  the  graph  when  some  of  the  linear  factors  are  imagi- 

nary     219 

132.  Relations  between  the  roots  and  the  coefficients  of  an  alge- 

braic equation 219 

133.  Symmetric  functions 221 

134.  Vanishing  and  infinite  roots 222 


CHAPTER  VI 

Fractional  Rational  Functions. 

135.  Definition  of  a  rational  function 226 

136.  Proper  and  improper  rational  fractions 226 

137.  Reduction  of  a  rational   function  to  its  lowest  terms       .        .  228 

138.  Zeros  of  a  rational  function 230 

139.  Poles  of  a  fractional  rational  function 230 

140.  Graph  of  a  fractional  rational  function 231 

141.  General  form  of  a  rational  function  in  terms  of  its  zeros  and 

poles 233 

142.  Partial  fractions 234 

143.  Resolution   into   partial  fractions,  when   the  poles  are   not 

simple 237 

144.  Modified  form  of  the  partial  fractions  in  the  case  of  imaginary 

poles 240 

145.  Fractional  rational  equations 241 

146.  Pressure  exerted  by  gases 244 


CHAPTER   VII 

Irrational  Functions. 

147.  Existence  of  irrational  functions 248 

148.  The  function  y  =  'Vx  and  its  principal  value    ....  250 

149.  The  graph  oiy  =  y/x 251 

150.  The  function  y  =  ^x^ .253 

151.  Properties  of  radicals 254 

152.  The  square  root  of  a  rational  function 256 

1.53.  Functions  which  involve  the  square  root  of  a  rational  function 

and  no  other  irrationality 258 

154.  Irrational  equations  of  the  simplest  type  .         .         .         .261 

155.  The  general  algebraic  function 262 


XVI 


CONTENTS 


CHAPTER   VIII 


Fractional  and  Negative  Exponents.     The  General  Power  Function. 
The  Exponential  Function  and  Logarithms. 


156.  Fractional,  negative,  and  vanishing  exponents 

157.  The  index  laws 

158.  The  principle  of  permanence    . 

159.  The  case  of  an  irrational  exponent 

160.  The  power  function  . 

161.  The  exponential  function 

162.  Graphs  of  exponential  functions 

163.  Properties  of  a-^ 

164.  Definition  of  logarithm 

165.  Graph  of  a  logarithmic  function 

166.  Properties  of  logarithms   . 

167.  Common  logarithms 

168.  Characteristic  and  mantissa 

169.  Properties  of  the  mantissa 

170.  Determination  of  the  characteristic 

171.  Arrangement  and  use  of  the  table  of  logarithms 

172.  Extraction  of  roots  by  means  of  logarithms 

173.  Logarithmic  calculations  which  involve  negative  number 

174.  Principles  used  in  logarithmic  calculations 

175.  Arrangement  of  the  calculation 

176.  The  logarithmic  or  Gunter  scale 

177.  The  slide  rule 

178.  The  general  notion  of  a  scale    . 

179.  Relation  between  the  logarithms  of  two  different  systems 

180.  Selection  of  a  standard  logarithmic  curve 

181.  The  derivative  of  the  logarithmic  function 

182.  The  numerical  value  of  e 

183.  Exponential  equations 

184.  The  calculation  of  a  table  of  logarithms   . 

185.  Applications  of  logarithms 

186.  Simple  interest       .  

187.  Compound  interest 

188.  Annuity 

189.  Interest  compounded  more  than  once  annually 

190.  The  compound  interest  law       ..... 

191.  Dampened  vibrations         ...... 

192.  Variation  of  density  and  pressure  in  the  atmosphere 

193.  Transmission  of  light  by  imperfectly  transparent  media 

194.  Cooling  bodies 

195.  Semi-logarithmic  paper     ... 

196.  Logarithmic  paper 


CONTENTS  xvii 


CHAPTER  IX 


Linear  Functions  of  More  than  One  Variable.     Linear  Equations  and 
Determinants  of  the  Second  and  Third  Order. 


PAGE 

319 
320 
321 
322 


197.  Functions  of  two  variables 

198.  Linear  functions  of  two  variables     ..... 

199.  Linear  equations        ........ 

200.  Simultaneous  linear  equations  ...... 

201.  General  foruiul;?  for  the  solution  of  two  simultaneous  linear 

equations  with  two  unknowns    ......     324 

202.  Determinants  of  the  second  order 325 

203.  Homogeneous  linear  equations  with  two  unknowns  .         .     326 

204.  Discussion  of  the  solutions  of  two  linear  equations  with  two 

unknowns 329 

205.  Properties  of  determinants  of  the  second  order        .        .        .     331 
20(5.    Determinants  of  the  third  order 332 

207.  Cof actors 337 

208.  The  principal  properties  of  determinants  of  tlie  tiiird  order     .     338 

209.  Solution  of  a  system  of  three  simultaneous  linear  equations 

with  three  unknowns  .......     342 

210.  Homogeneous  equations 344 

211.  An  application  of  linear  equations  in  Chemistry      .        .        .     346 

212.  Generalization  to    systems    of    n   linear    equations  with   n 

unknowns 350 


CHAPTER   X 

Permutations  and  Combinations. 

213.  The  notion  of  order 352 

214.  Permutations 352 

215.  The  number  of  permutations  of  n  elements  taken  k  at  a  time  354 

216.  Circular  arrangements 356 

217.  Permutations  when  all  of  the  elements  are  not  distinct  .         .  357 

218.  Two  classes  of  permutations 359 

219.  Combinations 361 

220.  Independent  combinations 363 

221.  The  binomial  theorem 364 

222.  Total  number  of  combinations 365 


CHAPTER  XI 
Probability. 

223.  Definition  of  probability    .         .         .       • 366 

224.  Compound  events 368 


xviii  CONTENTS 

PAGE 

225.  Repeated  trials 372 

226.  Application  to  life  insurance 373 

227.  Other  applications  of  the  theory  of  probability        .        .        •    376 


CHAPTER  XII 

Determinants  of  the  nth  Order  and  Systems  of  Linear  Equations  with 
n  Unknowns. 

228.  Definition  of  a  determinant  of  the  «th  order    ....     376 

229.  Another  method  for  determining  the  sign  of  a  term  of  the 

determinant 378 

230.  Properties  of  determinants 380 

231.  Minors 382 

232.  Cofactors 384 

233.  Solution  of  a  system  of  n  linear  eciuatioiis  for  n  unknowns     .  386 

234.  Homogeneous  equations 387 

235.  Systems  of  linear  equations  with  more  equations   than  un- 

knowns      ..........     388 

236.  Systems  of  linear  equations  with  fewer  equations  than  un- 

knowns      ..........     388 

237.  Application  of  determinants  to  the  theory  of  elimination        .    388 


CHAPTER   XIII 

Quadratic  Functions  of  Two  Independent  Variables  and  Simultaneous 
Quadratic  Equations. 

238.  Integral  rational  functions  of  two  independent  variables         .  391 

239.  Quadratic  function  of  x  and  y 392 

240.  Composite  and  non-composite  quadratic  functions  .         .         .  392 

241.  The  values  of  a  quadratic  function   ......  398 

242.  The  existence  of  solutions  of  a  quadratic  equation   .         .         .  399 

243.  Graph  of   a  function  defined  by  a  quadratic    equation  in  x 

and  y 401 

244.  Solution  of  a  system  of  simultaneous  equations  one  of  which 

is  linear  and  one  of  which  is  quadratic      ....  407 

245.  Simultaneous  quadratics 409 

246.  Equivalent  systems  of  simultaneous  equations          .        .        .  409 

247.  Normalization 411 

248.  Existence  of  four  solutions 414 

249.  Special  cases  of  two  simultaneous  quadratics           .         •         •  416 

250.  Case  I.     H  =  If  =  F  =  F'  -  G  =G'  =  0.     Neither  equation 

contains  a  first  degree  term  or  a  term  inxy      .        ■        ■  416 

251.  Case  II.     F=F'^G=G'  =  0 417 


CONTENTS 


XIX 


PAGE 

252.  Case  III.  Both  equations  contains  x  and  y  in  symmetric 
fashion,  so  that  the  equation  is  left  uualtered  if  x  and  y 
are  interchanged 420 

263.  Case  IV.  When  at  least  one  of  the  given  equations  is  com- 
posite     ........■••     423 

254.  A  new  method   for  the  general  case   of  simultaneous  quad- 

ratics .......••••    423 

255.  The  method  of  small  corrections 425 

256.  Applications  which  involve  simultaneous  quadratics        .        ,    427 


CHAPTER  XIV 

Sequences  and  Series  with  a  Finite  Number  of  Terms. 

257.  Continnous  and  discontinuous  variation           ....  428 

258.  Definition  of  a  sequence 429 

259.  Higher  progressions 430 

260.  Geometric  progressions 432 

261.  Series 433 

262.  Summation  of  series  by  mathematical  induction       .        .        .  433 

263.  General    characteristics    of    the    method    of    mathematical 

induction     ..........  434 

264.  The  summation  sign 437 

265.  Summation  of  a  series  whose  ^•th  term  is  an  integral  rational 

function  of  fc 438 

266.  Summation  of  some  other  simple  series 441 


CHAPTER   XV 
Limits. 

267.  Limits  suggested  by  series 443 

268.  Definition  of  a  limit 444 

269.  Infinity 447 

270.  Infinitesimals 448 

271.  Variables  which  remain  finite 448 

272.  A  theorem  about  infinitesimals  ......  449 

273.  Theorems  about  limits 450 

274.  Limit  of  a  quotient  of  two  variables 451 

275.  Limit  of  the  nth  power  of  a  positive  number  as   n  grows 

beyond  bound 454 

276.  Continuity  of  a  function •  455 

277.  Continuity  of  a  fractional  rational  function     ....  458 

278.  Indeterminate  forms 460 


XX 


CONTENTS 


CHAPTER  XVI 


Infinite  Series. 


279.  Non-terminating  geometric  progressions  . 

280.  Some  other  non-terminating  series   . 

281.  Convergence  and  divergence  of  infinite  series 

282.  Fundamental  criteria  for  convergence 

283.  Series  all  of  whose  terms  are  positive 
284-  Comparison  tests       ..... 
2^5.  Some  convenient  comparison  series  . 
28(5.  Ratio  test 

287.  Ratio  of  corresponding  terms  of  two  series 

288.  Series  with  positive  and  negative  terms    . 

289.  Conditionally  convergent  series 

290.  Alternating  series 

291.  Series  whose  terms  are  functions  of  X 

292.  Power  series       ...... 

293.  Equality  of  two  power  series    . 

294.  Expansion  of  functions  as  power  series    . 

295.  Expansion  of  rational  functions 

296.  Expansion  of  some  ii'rational  functions    . 

297.  The  expansion  of  (1  -f  a;)" 

298.  Exponential  series 

299.  Logarithmic  series 


PAGB 

465 
465 
466 
468 
471 
471 
472 
474 
479 
482 
484 
484 
485 
486 
488 
488 
489 
492 
493 
495 
496 


APPENDIX 

Table  1 .     Four  Place  Logarithms  of  Numbers 
Table  2.     American  Experience  Table  of  Mortality 


498 
500 


COLLEGE   ALGEBRA   WITH 
APPLICATIONS 

CHAPTER   I 

THE  NUMBER  SYSTEM  OF  ALGEBRA 

1.  The  positive  integer.  The  most  fundamental  notion  of 
arithmetic,  that  of  tlie  positive  integer^  is  obtained  by  the 
process  of  counting  and  becomes  familiar  to  us  in  our  early 
childhood.  The  names  and  symbols  used  for  the  positive 
integers  are  well  known  to  all.  The  following  fact  is  also 
familiar,  but  its  importance  justifies  an  explicit  formulation 
at  this  time.  There  exists  a  first  positive  integer,  namely 
unitg,  but  there  is  no  last. 

2.  Addition,  subtraction,  and  multiplication  of  positive  in- 
tegers. If  the  letters  a,  5,  c,  etc.,  are  used  to  denote  given 
positive  integers,  we  know  from  our  early  training  in  arith- 
metic how  to  form  their  sums,  differences,  and  products.  In 
fact,  we  know  that  the  sum,  a  +  5,  and  the  product,  a  x  b  or 
ab,  is  always  again  a  positive  integer.  This  will  also  be 
true  of  the  difference,  a  —  b,  provided  that  a  is  greater  than 
6,  that  is,  (a  >  b). 

The  following  laws  sum  up  the  most  essential  properties 
of  positive  integers,  and  may  be  regarded  as  justified  by  our 
experience  with  a  large  number  of  special  cases : 

I.  If  we  add  a  positive  integer  b  to  another  positive  integer 
a,  we  always  obtain  a  uniquely  determined  positive  integer 

c=  a  -\-  b, 

which  is  called  the  sum  of  a  and  b. 

II.  Addition  is  commutative;  that  is,  a  +  b  =  b  +  a. 

1 


2  THE   NUMBER   SYSTEM   OF   ALGEBRA  [Art.  2 

III.  Addition  is  associative ;  that  is, 

a+(b  +  c)  =  (a  +  J)  +  c. 

IV.  Addition  is  monotonic ;  that  is,  if  a  is  greater  than  b, 
then  a  +  0  is  greater  than  b  +  c.     In  syynbols, 

if  a  >  b,  then  a  +  c  >  b  -\-  c. 

V.  If  zve  midtiply  a  positive  integer  a  by  another  positive 
integer  b,  tve  alivays  obtain  a  uniquely  determined  positive 
^'^^^9er  c  =  axb  =  a.  b  =  ab, 

tvhich  is  called  the  product  of  a  by  b. 

VI.  Multiplication  is  commutative ;  that  is,  a  x  b  =  b  x  a. 

VII.  Multiplication  is  associative;  that  is, 

a  x(b  X  c')  =  (a  X  b')  X  c. 

VIII.  Multiplication  is  monotonic;  that  is, 

if  a  >  b,  then  a  x  c>  b  x  c. 

IX.  Multiplicatioyi  is  distributive  with  respect  to  addition; 
^^"<'^''^'  c(a+6)  =  ca  +  6'5. 

Subtraction  may  be  clelined  as  follows.  If  a  and  b  are 
positive  integers,  such  that  a  >  b,  then  there  exists  a  positive 
integer  x,  tvhich  added  to  b  will  give  a  as  a  sum;  that  is,  if 
a  ">  b,  there  exists  a  positive  integer  xfor  which 

b  +  x=  a. 

This  integer  x  is  denoted  by 

x=  a  —  b, 

and  is  called  the  difference  between  a  and  b.     Moreover,  a  and 
b  are  called  minuend  and  subtrahend  respectively. 

The  laws  of  subtraction,  which  are  familiar,  may  be  derived 
from  this  definition  and  the  laws  I  .   .   .  IX. 

In  Algebra  we  wish  to  make  general  statements,  applicable  to  all 
integers.  While  we  can  easily  verify  that  the  above  laws  are  true  of  a 
great  mnn)j  integers,  all  of  those  less  than  ten,  for  instance,  we  cannot,  by 
the  method  of  actual  test,  assure  ourselves  that  they  are  true  of  all 


Akt.  3]         FACTORS   OR   DIVISORS   OF   INTEGERS  3 

integers.  For  we  should  never  get  through  with  the  task  of  testing  them 
all,  since  there  is  no  last  integer.  (See  Art.  1.)  Thus  it  is  impossible  to 
justifij  the  general  valididj  of  the  above  laws  on  the  basis  of  experience  alone. 
On  the  other  hand,  a  lofjicul  proof  of  these  laws  could,  at  best,  be  only 
partially  successful.  For  sucli  a  logical  proof  would  consist  in  showing 
that  they  are  necessary  consequences  of  certain  other  law's,  and  then  the 
latter  would  have  to  remain  without  a  strictly  logical  proof,  however 
plausible  it  might  seem  that  they  should  be  true. 

Since  the  laics  I  .  .  .  IX  can  never  be  tested  completely  by  experience,  and 
since  they  can  never  receive  a  complete  proof  by  the  method  of  loyic,  any 
statement  that  these  laws  are  universally  true  is  of  thr  nuturr-  of  an  assumption. 
We  therefore  speak  of  these  laws  as  fundamental  assumptions  *  of  Algebra. 
They  are  called  fundamental  for  two  reasons.  First,  because  they  are 
used  in  all  of  the  elementary  operations  of  Arithmetic,  and  second, 
because  it  can  be  shown  that  they  are  of  the  greatest  importance 
throughout  all  of  Algebra,  as  will  become  apparent  very  soon.  To  il- 
lustrate the  first  point,  let  us  multiply  8  by  13.     We  reason  as  follows  : 

8  X  13  =  8(10  +  3)  =  8  •  10  +  8  •  3  =  80  +  21 

=  80  +  (20  +  4)  =  (80  +  20)  +  4  =  100  +  4  =  104, 

where  we  have  used  laws  V,  IX,  III,  I. 

This  example  illustrates  the  following  general  statement:  To  perform 
a  calculation  with  positive  integers  ice  need  to  know,  in  the  first  place,  the 
results  obtained  by  the  addition,  subtraction,  and  multiplication  of  any  two 
integers  less  than  ten  (the  addition  and  multiplication  tables^;  and  in  the 
second  place,  we  must  know  how  to  apply  the  nine  fundamental  laws. 

3.  Factors  or  divisors  of  integers.  Since  some  positive 
integers  may  be  obtained  by  multiplying  together  two  or 
more  others,  it  becomes  important  to  understand  the  fol- 
lowing definitions  : 

If  a  positive  integer  n  can  he  expressed  as  a  product  of  two 
or  more  positive  integers.,  each  of  the  latter  integers  is  said  to  he 
a  factor  or  divisor  of  n. 

If  an  integer  has  no  integral  divisors  excepting  itself  and 
unity.,  it  is  said  to  he  a  prime  number.  All  other  integers  are 
said  to  he  composite. 

The  lowest  prime  numbers  are  2,  3,  5,  7,  11,  13,  17,  19. 

*  We  do  this  in  spite  of  the  fact  tliat  it  is  possible  to  build  up  Algebra  on 
fewer  assumptions  than  we  have  here  listed. 


4  THE    NUMBER   SYSTEM   OF   ALGEBRA  [Art.  4 

4.  Division  of  integers.  Let  B  and  d  be  two  positive  in- 
tegers, D  being  greater  than  d.  \i  D  has  c?  as  a  factor,  we 
may  write 

(1)  D  =  dq, 

where  q,  the  second  factor,  is  also  a  positive  integer.     It  is 
customary  to  write  (1)  in  the  equivalent  form 

(2)  .  =  f, 

and  to  speak  of  q  as  the  quotient  obtained  by  dividing  D  (the 
dividend)  by  d  (the  divisor). 

Thus,  we  may  write  14  =  7  •  2  or  2  =  ^.  In  this  case  Z)  =  14,  rf  =  7, 
q  =  2. 

But  suppose  that  d  is  not  a  factor  of  2>,  and  let  us  ex- 
amine the  successive  multiples  d,2d,  3  c?,  etc.,  of  d.  Let  qd 
be  the  largest  integral  multiple  of  d  which  is  less  than  Z>, 
so  that  the  next  multiple  (jq  -^l^d  will  be  greater  than  D. 
We  shall  then  have 

qd<D,   (^q  +  \~)d>D, 

and  the  difference  D  —  qd  will  be  a  positive  integer  r  which 
is  less  than  d  ;  that  is, 

(3)  D  —  qd  =  r,  or  D  =  qd  -\- r,  r<d. 

The  two  positive  integers,  q  and  r,  which  are  obtained 
from  I)  and  d  by  this  process  are  called  the  quotient  and  the 
remainder  respectively.  The  process  itself  is  called  division, 
and  the  numbers  D  and  d  are  the  dividend  and  divisor 
respectively. 

Thus,  li  D  —  60,  and  d  =  7,  we  find  60  =  8  •  7  +  4,  so  that  the  quotient 
is  7  =  y  and  the  remainder  r  =  4. 

We  observe  that  the  case,  where  f?  is  a  factor  of  i),  is 
included  as  a  special  case  under  (3)  ;  it  arises  when  the  re- 
mainder r  is  equal  to  zero. 


Art.  5]  THE   GREATEST   COMMON   DIVISOR  5 

5.  The  greatest  common  divisor  of  two  integers.  As  in 
Art.  4  let  D  and  d  be  two  positive  integers  and  let  D  be  the 
greater  of  the  two.  Let  q  be  the  quotient  and  r  the  re- 
mainder obtained  when  we  divide  D  by  d.  We  shall  then 
have 

(1)  D  —  qd  =  r  or  I)  =  qd  +  r,  r  <  d. 

If  r  =  0,  d  is  a  divisor  of  D,  and  since  d  is  clearly  its  own 
greatest  divisor,  d  will  then  be  the  greatest  common  divisor 
oi  D  and  d. 

But  suppose  that  r  is  not  equal  to  zero.  Since  q  is  an 
integer,  equation  (1)  shows  that  any  common  divisor  of  JJ 
and  d  will  also  be  a  divisor  of  r,  and  therefore  a  common 
divisor  of  d  and  r.  More  specifically,  the  greatest  common 
divisor  of  D  and  d  will  also  be  the  greatest  common  divisor 
of  d  and  r. 

If  r  is  itself  a  divisor  of  d,  it  will  be  the  greatest  common 
divisor  of  r  and  d,  and  therefore  also  of  d  and  D.  If  r  is 
not  a  divisor  of  c?,  let  us  divide  t?  by  r  Qd  being  greater  than 
r  on  account  of  (1)),  and  let  q'  be  the  quotient  and  r'  the 
remainder,  so  that 

(2)  d  —  q'r  =  r'  or  d  =  q'r  +  r',  r'  <  r. 

Reasoning  just  as  above,  we  see  from  (2)  that  the  greatest 
common  divisor  of  d  and  r  will  also  be  the  greatest  common 
divisor  of  r  and  r' .  This  will  be  r'  itself  if  r'  is  an  exact 
divisor  of  r.  In  that  case  r'  will  also  be  the  greatest  com- 
mon divisor  of  D  and  d.  If,  however,  r'  is  not  an  exact 
divisor  of  r  we  continue  the  division  process  and  obtain 

(3)  r  -  q"r'  =  r"  or  r=  q"r'  +  r'\  r"  <  /. 

As  we  continue  in  this  way  tlie  successive  remainders  r,  r',  r". 
etc.  become  smaller  and  snuiUer.  We  shall  tinally  reach  a 
vanishing  remainder,  that  is,  we  shall  finally  obtain  an  exact 
division.  Our  argument  shows  that  the  last  no )t- vanishing 
remainder  obtained  hg  this  process  is  the  greatest  common  divisor 
of  D  and  d. 


6  THE   NUMBER   SYSTEM   OF   ALGEBRA  [Art.  6 

This  greatest  common  divisor  may  be  equal  to  1.  Since 
any  two  integers  have  unity  as  a  common  divisor,  it  is  cus- 
tomary to  ignore  unity  as  a  divisor  and  to  say  that  two  such 
numbers  have  no  common  divisor.  Two  numbers  whose  great- 
est common  divisor  is  unity,  are  said  to  he  relatively  prime. 

The  process  for  finding  the  greatest  conmion  divisor  of  two  positive 
integers  is  frequently  called  Euclid's  algorithm.  It  was  described  and 
applied  in  Euclid's  famous  treatise  on  Geometry  called  the  Elements. 

The  lowest  common  multiple  of  several  integers  is  the  smallest 
integer  which  is  exaetly  divisible  by  each  of  these  integers. 

6.  The  prime  factors  of  an  integer.  Let  n  be  any  positive 
integer,  and  let  us  write  down  all  of  the  prime  numbers  2,  3, 
5,  etc.  which  are  not  greater  tlian  n.  Then  we  can  find,  by 
the  process  of  Art.  5,  the  greatest  common  divisor  of  n  and 
the  first  prime  number  2.  Since  2  is  a  prime  number  its 
only  divisors  are  1  and  2,  so  that  the  greatest  common 
divisor  of  n  and  2  can  only  be  either  1  or  2.  If  it  is  1,  n  and 
2  are  prime  to  each  other.  If  it  is  2,  then  2  is  a  divisor  of  n 
and  we  may  write  n  =  2  n'  where  n'  is  again  a  positive 
integer. 

In  the  latter  case  we  may  examine  n'  in  the  same  fashion 
and  continue  in  this  way.     We  shall  finally  find 


where  r  and  m  are  integers,  and  where  m  is  either  equal  to 
unity  or  else  an  integer  not  divisible  b}^  2. 

We  now  examine  whether  m  is  divisible  by  the  next  prime 
number  3.     We  either  find  that  it  is  not,  or  else 

m  =  3*jt?, 

where  j[?  is  an  integer  not  divisible  by  either  2  or  3.  As  we 
proceed  in  this  way,  we  shall  finally  obtain  our  original 
number  7i  expressed  in  the  form  of  a  product 

(1)  w  =  2'3«5'».. 

decomposed  into  its  prime  factors. 


Art.  7]  GEOMETRIC   REPRESENTATION  7 

EXERCISE  I 

In  examples  1-0  divide  the  larger  integer  by  the  smaller  and  find  the 
quotient  and  remainder.  Use  the  result  to  write  the  dividend  in  the 
form  of  equation  (3),  Art.  4. 

1.  Divide  70  by  8.  4.   Divide  115  by  36. 

2.  Divide  64  by  11.  5.   Divide  365  by  30. 

3.  Divide  99  by  12.  6.    Divide  7284  by  63. 

7.  Find  the  greatest  common  divisor  of  56  and  49. 

8.  Find  the  greatest  common  divisor  of  663  and  247. 

9.  Decompose  504  into  its  prime  factors  and  ^Tite  the  result  in  the 
form  of  (1).  Art.  6. 

10.  Find  the  lowest  common  multiple  of  3,  6,  .5,  4. 

11.  Prove  the  theorem  :  if  p  and  q  are  two  positive  integers  and  if  d 
is  their  greatest  common  divisor,  then  their  lowest  common  multiple  is 
equal  to  pq/d. 

7.   Geometric  representation  of  positive  integers.     We  may 

adopt  a  line-segment  of  convenient  length  (say  one  inch)  as 
unit  of  length.  If  we  find  that  this  unit  is  contained  in 
another  line-segment  exactly  n  times,  we  say  that  the  length 
of  the  latter  is  ii  units.  We  may  also  say  that  the  two  line- 
segments  represent  the  integers  1  and  n  respectively.  For 
purposes  of  comparison  of  two  different  line-segments  it  will 
be  convenient  to  use  a  line  of  reference  with  a  scale  marked 
upon  it  in  the  following  way.     On  any  1234 

line  OX  (Fig.  1),  terminating  at  0  (which  o  a  b  c  b  '^ 
point  is  called  the  origin  of  the  scale),  ^^'^■^ 

but  unbounded  in  the  direction  toward  X,  we  lay  off  line- 
segments  OA,  AB,  BC,  etc.,  each  of  unit  length,  and  label 
the  points  A,  B,  (7,  Z),  etc.,  with  the  numbers  1,  2,  3,  4,  etc. 
We  thus  obtain  a  point  of  this  line  corresponding  to  every 
positive  integer  and  vice  versa.  Moreover  the  correspond- 
ence is  such  that  the  line-segment  which  joins  the  origin  to 
the  point  labeled  n  on  the  scale-  is  just  n  units  long.  We 
shall  usually  think  of  the  line-segments  OA,  OB,  00,  OB, 
etc.,  as  representing  the  numbers  1,  2,  3,  4,  etc.,  although 
other  line-segments  of  the  same  length  would  do  just  as 
well. 


8  THE   NUMBER   SYSTEM   OF   ALGEBRA  [Art.  8 

8.  Rational  numbers.  This  representation  shows  that 
there  are  many  points  on  the  line  OX  which  do  not  corre- 
spond to  any  integer.  We  now  introduce  fractions  so  as  to 
make  the  correspondence  between  the  points  of  the  scale 
and  the  numbers  more  complete.  Let  us  divide  the  line- 
segment  OA  (which  is  of  unit  length)  into  an  integral  num- 
ber of  equal  parts.  Let  there  be  q  such  parts  and  let  us 
agree  to  represent  the  length  of  one  of  these  parts  by  the 
ij  J    2    3    4  symbol  I/5'.      (See  Fig.  2.)     Let  us  then 

o  A   B  en  label  the   points  of  OX  whose  distances 

^^^«-  ~  from  0  are  equal  to  1/^,  2(l/g),  3(1/^), 

etc.,  by  the  symbols  1/q,  2/q,  S/q,  etc.,  and  let  us  do  this 
for  all  values  of  q  beginning  with  ^  =  2,  q  =  S,  5-  =  4,  etc. 
We  thus  obtain  a  very  large  number  of  intermediate  points 
on  the  scale,  each  of  which  will  have  a  symbol  of  the  form 
p/q  attached  to  it,  where  p  and  q  are  positive  integers.  We 
now  agree  further  that  each  of  these  new  symbols  shall  be 
regarded  as  a  number  and  that  these  new  numbers  shall  be 
called  fractions,  p  is  called  the  numerator  and  q  the  denomi- 
nator of  the  fraction  p/q.  Of  course,  unless  5'  =  1,  these  new 
numbers  are  not  integers.  Litegers  and  fractions  taken  to- 
gether are  called  rational  numbers. 

TJie  fraction  p/q  is  the  measure  of  the  length  of  the  line- 
segment  which  joins  the  origin  to  the  point  labeled  p/q  on 
the  scale.,  this  length  being  measured  in  terms  of  the  line- 
segment  OA  as  unit  of  length.,  where  OA  is  the  line-segment 
ivhich  joins  the  origin  to  the  point  of  the  scale  which  is 
labeled  1. 

It  follows  from  this  definition  that  if  a  line-segment  p 
units  long  (p  being  an  integer)  is  divided  into  q  equal 
parts,  the  length  of  each  part  will  be  p  qtlia  of  a  unit ;  and 
also  that 

(1)  P+P'  -P_^.p1. 

We  now  introduce  a  new  kind  of  division,  different  from 
that  of  Art.  4,  by  the  following  definition: 


Akt.  9]     SOME   PROPERTIES   OF   RATIONAL   NUMBERS         9 

To  divide  a  jwsifive  integer  D  {the  dividend)  hy  another 
positive  integer  d  {the  divisor)  means  to  divide  the  line- 
segment  of  letigth  D  units  into  d  equal  parts  and  to  find  the 
length  q  of  one  of  these  parts.  This  length  q  in  called  the 
quotient. 

We  now  know  that   this  problem  always  lias  a  solution, 

namely,  2> 

q  =  D  divided  hy  d  =  the  fraction—  • 

Consequently  the  symbol  for  a  fraction  D/d  may  be  read  : 
D  divided  hy  d.  Since  we  have  attached  a  concrete  meaning 
to  the  operation  of  dividing  one  integer  by  another,  the  fol- 
lowing definition  is  now  intelligible. 

Any  number  which  may  he  regarded  as  the  quotient  of  one 
positive  integer  divided  by  another  is  called  a  j^ositive  rational 
number. 

Of  course  all  integers  are  included  among  the  rational  nurnl)ers,  since 
any  integer  may  be  regarded  as  being  the  quotient  obtained  by  dividing 
that  integer  itself  by  unity. 

If  D  and  d  are  any  two  positive  integers,  we  may  therefore 
always  write 

(2)  I)-^d  =  ^=q 

where  q  is  either  an  integer  or  a  fraction,  and  this  relation 
may  be  regarded  as  equivalent  to 

(3)  D  =  dq. 

9.  Some  properties  of  rational  numbers.  It  is  clear  that 
the  process  of  introducing  intermediate  points  on  the  scale 
by  making  use  of  fractions  will  enable  us  to  label  many  of 
these  points  in  more  than  one  way.  Thus,  the  point  which 
corresponds  to  the  symbol  ^  is  the  same  as  that  which  corre- 
sponds to  |,  and  the  lengths  of  the  corresponding  line-seg- 
ments are  equal.  We  therefore  agree  that  the  fractions  | 
and  I  shall  be  regarded  as  equal.  With  this  introduction, 
the  following  statements  will  be  easily  understood. 


10  THE   NUMBER   SYSTEM  OF   ALGEBRA  [Art.  9 

(a)  A  rational  number  p/q^  where  p  and  q  are  integers,  is 
said  to  he  in  its  lowest  terms,  if  the  integers  p  and  q  have  no 
common  divisor  except  unity,  or  in  other  words,  if  p  and  q  are 
relatively  prime. 

(5)  Any  rational  number  may  be  reduced  to  its  loivest  terms 
by  dividing  both  numerator  and  denotninator  by  their  greatest 
common  divisor. 

(c)  Two  rational  nu^nbers  are  equal  to  each  other  if  they  re- 
duce to  the  same  number  when  both  are  expressed  in  their  lowest 
terms. 

Thus  we  have  -t-  =  -J-' 

q      mq 

More  gCDerally,  if  p' /q'  is  equal  to  p" /q",  both  of  these  frac- 
tions will  reduce  to  the  same  fraction  p/q  when  written  in 
their  lowest  terms.      We  shall  therefore  have 

p'=m'p,  q'  =  m'q,  p"  =  m"p,  q"  =m"q, 


where  m'  and  m"  are  integers,  and  consequently 
whence 


p'q"  =  m'm"pq,  p"q'  =  m'm"pq, 


(1)  p'q"=p"q'. 

This  relation  must  hold  if  the  fractions  p' /q'  and  p" /q"  are 
equal. 

Conversely,  if  (1)  is  satisfied,  we  may  obtain  from  (1), 
by  dividing  both  of  its  members  by  q'q",  the  equation 

pi  _p" 
q'      q" 

We  may  therefore  replace  statement  (c)  which  involves 
the  definition  of  equality  of  two  rational  numbers,  by  the 
following : 

(c7)    Ttvo  rational  numbers  p' /q'  and  p" /q"  are  equal,  if 

and  only  if  ,  ,,        ,,  , 

^  •'  p  q    =p"q' . 

If  two  rational  numbers,  p/q  and  p'  /q',  are  not  equal,  the 
points,  P  and  P',  which  represent  them  on  the  scale  of 


AuT.  10]  ADDITION   AND   SUBTRACTION  11 

rational  numbers,  will  not  coincide.     We  shall  say  that 
^  is  less  than  ^  ,    or  ^  precedes  ^  , 

if  P  is  closer  to  the  origin  than  P' .    (See     ,12     p/g  p/g' 

Fisr.  3.)     In  symbols  we  express  this  rela-    o  a  b     p    p' 
,.  ''     ,   ^         .,/  ^  Fig.  3 

tion  by  writing  , 

q      q' 

The  equivalent  relation      -,  > - 

may  be  read  jo'/^'. is  greater  i\vA\\p/q^  ov  p' /q'  follows  jo/^'. 

We  easily  recognize  the  validity  of  the  following  numerical 
test : 

(e)  The  rational  number  p/q  is  greater  than  p' /q' ^  if  and 
only  if  the  integers  p,  q^  p',  q'  are  such  that 

pq'>p'q. 

Similarly,  p/q  is  less  than  p' /q',  if  and  only  if 
pq'  <p'q. 

Thus  I  >  I  since  3  •  3  >  2  •  4.  To  prove  statement  (e)  it  suflBces  to  re- 
duce the  two  fractions  which  are  to  be  compared  to  a  common  denomina- 
tor qq'.     Thus  we  have  f  =  t\  and  f  =  r\,  so  that  J  >  f  because  9  >  8. 

(/)  The  tests  given  under  ((i)  and  (*')  alivays  enable  us  to 
decide  whether  two  given  positive  rational  numbers  are  equal  or 
not,  and,  in  the  latter  case,  to  decide  ivhich  is  the  greater. 

{g}  If  no  tivo  of  the  three  rational  numbers  p/q,  p'  /q'-,  p"  /q^' 
are  equal,  and  if 

either  p  /  q  <::  p'  /  q'  <p"  /q"  or  p/q>  p'  /q'  >  p"  /q" -, 

then  2^' /q'  is  said  to  be  between  p/q  and  p" /q". 

10.  Addition  and  subtraction  of  rational  numbers.  Let 
a— p/q  be  any  rational  number  and  let  A  be  the  corre- 
sponding point  of  the  number  scale.  (See  Fig.  4.)  Then 
OA  is  a  units  long  and  we  shall  call  0  the  origin  and  A  the 


12  THE   NUMBER   SYSTEM   OF   ALGEBRA        [Art.  10 

terminus  of    this  line-segment    OA.     Let  a'=p'/q'   be   any 
other  rational  number  represented  by  the  line-segment  OA', 

whose    origin    is    0    and    whose 
k — -—a^a-i-a  ^^^         >,  terminus  is  A' .      We  now  define 


5  li     A'  'Y'  addition  as  follows. 

To  add  a'  to  a  ive  first  place  the 
origin  of  the  line-segment  OA,  which  represents  a,  upon  the 
origin  of  the  number  scale.  We  then  place  the  origin  of  the 
line-segment  OA',  which  represents  a',  7ipon  A,  the  terminus  of 
OA.  Let  A"  he  the  position  which  the  terminus  of  OA'  will 
then  occupy.  The  liiie-segment  OA"  will  then  represent  the 
sum  a  +  a' . 

The  arithmetic  rule  for  forming  the  sum  of  the  two 
rational  numbers,  ^/^  and  p'/q',  follows  from  this  definition. 
On  account  of  statement  (c)  of  Art.  9,  we  may  write 

(1)  a=E  =  ^,   a'  =  4  =  ^' 

q       qq'  q'       qq' 

thus  reducing  the  two  fractions  to  a  common  denominator 
qq' .  If  we  divide  our  original  unit  of  length  into  qq'  equal 
parts,  equations  (1)  tell  us  that  a  con.tains  pq'  and  that  a' 
contains  p'q  of  these  smaller  units.  Thus  their  sum  con- 
tains jo^-' +jt?' 5-  of  these  smaller  units,  so  that 

(2)  p_^p[^pq'  +p'q^ 
q       q'  qq' 

since  each  of  the  smaller  units  is  equal  to  the  original  unit 
divided  by  qq'. 

On  the  basis  of  the  above  definition  for  addition,  or  of  the 
equivalent  formula  (2),  we  can  now  easily  verify  that  the 
fundamental  laws  /,  //,  ///,  IV  of  Art.  2  are  true,  not  merely 
when  the  symbols  there  used  stand  for  positive  integers,  but  also 
if  these  symbols  represent  any  positive  rational  numbers. 

The  geometric  construction  for  subtraction  and  the  corre- 
sponding formula  are  so  immediate  as  not  to  call  for  a 
separate  discussion. 


Arts.  11,12]  MULTIPLICATION  OF   RATIONAL  NUMBERS   13 

11,  Multiplication  of  rational  numbers.  To  multiply  a 
fraction  'pjq  by  a  positive  integer  jo'  means  merely  to  take 
the  fraction  ip'  times,  so  that 


'  ^P^P'Z. 


P 


To  multiply  p/(i  by  l/q  means  the  same  thing  as  to  divide 
p/q  by  q\  that  is,  to  divide  the  line-segment  of  length  p/q 
into  q'  equal  parts  and  to  take  one  of  these  parts.     Thus 

q'       q       q'q 

We  combine  these  two  special  cases  into  the  following 
definition  : 

To  multiply  a  rational  number  p/q  {the  multiplicand')  hy  the 
rational  number  p' /q'  (the  multiplier)  means  to  find  a  third 
rational  number  p"/q"  (the  product)  such  that 

(1)  p1  =  pL^p^i^. 

q         q        q       q  q 

In  other  words,  multiplication  of  two  fractions  is  carried  out 
by  multiplying  the  two  numerators  and  the  two  denom- 
inators. 

It  is  easy  to  show,  as  a  consequence  of  this  definition,  that 
the  laws  V,  VI,  VII,  VIII,  IX  of  Art.  2  will  be  true  for 
positive  rational  numbers.  If  we  combine  this  result  with 
the  statement  at  the  end  of  Art.  10,  we  see  that  the  nine 
fundamental  laws  of  Art.  2  apply  not  only  to  positive  integers 
but  also  to  all  positive  rational  numbers. 

12.  Geometric  construction  for  the  product  of  two  posi- 
tive rational  numbers.  The  following  familiar  construction 
enables  us  to  find  the  product  of  two  rational  numbers  as  a 
line-segment  when  the  numbers  themselves  are  given  as 
line-segments. 

On  OX  (Fig.  5)  lay  oE  OA  =  a  units.  Through  0  draw 
any  line  OY,  not  coinciding  with   OX,  and  on  OY  lay  o£E 


14 


THE   NUMBER   SYSTEM   OF   ALGEBRA        [Art.  13 


0U=  1  unit  and  OA'  =  a'  units.     Join  U  to  A  and  draw  a 
line  parallel  to  AU  through  A'.     Let  P  be  the  point  in 

which  the  latter  line  intersects 
OX.  Then  OP  will  contain  aa' 
units,  so  that  OP  will  represent 
the  product  aa'. 

In  fact,  the  triangles  OAU  and 
OPA'  are  similar,  so  that 


Fig.  5 


OP:  0A=  OA':OU, 
OP:a  =  a':l, 
OP  =  aa'. 


or 
whence, 

This  construction  becomes  especially  convenient  when  OY 
is  taken  at  right  angles  to  OX. 


13.  Division  of  positive  rational  numbers.  According  to 
Art.  11,  the  product  of  two  rational  numbers  is  again  a 
rational  number.  The  problem  of  finding  one  of  these 
factors,  when  the  other  factor  and  the  product  are  given, 
constitutes  division.  Let  d  =  p/q  be  the  given  factor 
(divisor)  and  let  I)  =  p'/q'  be  the  given  product  (dividend). 
We  wish  to  find  a  rational  number  x  =  p"/q"  such  that 


(1) 


E 


q  q"        q' 


We  may  reduce 


^and^ 
qq"  q 


to  a  common  denominator,  writing 


Jr.Ji 


£PL  =  PJJEL  and  £-  =  ^^l^il, 
qq'       ([qq  q        qqq 


so  that  (1)  becomes 


pq'p"  _  p'qq" 


qq'  q" 


qq'q" 


ART.  13]  DIVISION   OF   RATIONAL   NUMBERS  15 

But  these  two  fractions,  with  equal   denominators,  can   be 
equal  only  if  their  numerators  are  equal,  that  is,  if 

(2)  (pq')p"=(p'q)q"- 

But,  according  to  Statement  (c?),  Art.  9,  equation  (2)  implies 
that  the  fractions  j»"/r^"  and  p'q/pq'  are  equal.     Therefore 

(3)  a:  =  P^  =  P^  =  ^xi. 

q        pq       q       p 

We  have  obtained  the  familiar  rule  for  division  of  fractions, 

(4)  pL^p^pL^i, 

q'       q       q'       p 

which  is  usually  expressed  as  follows :  To  divide  hy  the 
fraction  p/q  is  equivalent  to  multiplyijig  hy  the  reciprocal  frac- 
tion q/p.  For  two  fractions,  such 
as  p/q  and  q/p,  whose  product 
is  equal  to  unity  are  said  to  be 
reciprocal  to  each  other. 

It  is  easy  to  give  a  geometri- 
cal construction  for  division.  To 
divide  a  by  a'  we  lay  off  (see  Fig. 
6)  OA  =  a  units,  OA'  —■  a'  units, 
0U=1  unit.  We  then  join  A  to  A'  and  through  C/"  draw  a 
line  parallel  to  AA',  intersecting  OX  in  P.     Then 

OP  =  x  =  —  units. 

a 

In    fact,   the  similar    triangles    OPU  and  OAA'   yield   the 
proportion  OP^OA        ,  OP  ^  a. 

OU      OA''  °'     1        a'' 

whence  follows  OP  =  —  • 

a 

This  construction,  as  well  as  our  arithmetical  argument, 
shows  that  division  is  unique;  that  is,  two  positive  rational 
numbers,  D  and  d,  determine  a  unic^ue  rational  number  as 
their  quotient. 


16  THE   NUMBER   SYSTEM   OF   ALGEBRA        [Art.  U 

EXERCISE  II 
Perform  the  following  indicated  operations  arithmetically  and  geo- 
metrically. 

1.    1  +  ^.  3.    fxf.  5.    (i-i)^(i-2). 

6 

2«  —  1  4-I-  —  i.  'oii' 

•      9  3-  ^-      6     •     3-  ^  +  i 

Reduce  the  following  expressions  to  simpler  forms  and  state  which  of 
the  nine  laws  of  Art.  2  you  are  using. 

7.  (6o  +  3  i  +  5/)5  r/.  11.  (3  ac  +  ade  +  uf+  a)  ^  «. 

8.  (a  +  b)(c  +  (1).  12.  (ab  +  ac)^{b  +  c). 

9.  (a  +  b  +  c)(d  +  e+f).  13.  (ac +  hc  + ad  +  bd) -^  (a+ b). 
10.  (x  +  a)(x  +  b)(x+  c).  14.  (xx  +  2  xy  +  yy)  ^  (x  +  y). 

Reduce  the  following  expressions  to  the  form  of  simple  fractions, 
11  1,1 

«       ^  21.  ^-Jx-  +  ^^+l. 

p^r       qs      ps  +  rq 

16.^4-^-  '?       ' 

7      *■  1+11  +  1 

P2    ^ ?!  _£ / 

17.  ^  +  ^  +  ^.  r-T  •  1  +  1" 

11  23.  i • 


18.   -  X  - 


b  1  + 

1  + 


19.   1±J!^      '^ 


1  +c 

^+7  24.  ^ 

1  + 


20.  ^L±i^l±/  "    ■    l+x  +  ^^ 

c  +  d      g  +  h  1  +  X 

25.    Arrange  the  following  fractions  in   order  of  magnitude  : 


14.   A  further  property  of  the  rational  numbers.    Let  a  =p/q 

and    a'  =  j)' /q'    be    two    rational    numbers    and    let    a  <  a'. 

^Moreover,  let  OA   and   OA'  (Ficr.  7)  be  the 

I III, X  .  V       c^         / 

o       ABA'  corresponding  line-segments  on  OX,  so  that 

(1)  OA  =  a=^,    OA'  =  a'=^,   AA' =  a' -  a. 

q  q' 


Akt.  15]     THE   EXISTENCE  OF   IRRATIONAL  NUMBERS      17 

Let  B  be  the  point  which  bisects  AA'.     Then 
0B=  OA+IAA'  =  a  +  i(a'  -  a) 

or,  if  we  denote  by  h  the  length  of  OB, 
(2)  OB=h=l{a+a'^. 

But  by  (1)  we  find  that 

2  -iKq       q'J  2qq' 

is  again  a  rational  number,  and  it  is  obviously  between  a  and 
a' .     In  the  same  way  we  may  find  another  rational  number 

between  a  and  h,  still  another  between  a  and  5',  etc.  Thus, 
hetiveen  any  two  rational  numbers,  no  matter  hoiv  close  together 
they  may  be,  there  are  always  infinitely  many  other  rational 
numbers. 

Therefore,  the  rational  numbers  give  rise  to  infinitely 
many  points  infinitely  close  together  on  the  number  scale  of 
OX,  and  in  any  interval  on  OX  there  lie  infinitely  many  points 
(^called  the  rational  points  of  the  scale^  ivhose  distances  from  0 
are  represented  by  rational  numbers.  This  is  often  expressed 
by  saying  that  the  rational  points  of  the  scale  form  a  dense  set. 

15.   The   existence  of   irrational  numbers.      It  is  not  true, 
however,  tliat  the  distance  from  0  to  every  point  of  OX  can 
be  represented  by  a  rational  num- 
ber.      To   show  this,   let   us   con- 
struct a  square  (Fig.  8)  on  OA  as 
a  side,  where   OA  =  1    unit,   and      6        a   b 
draw  the  diagonal  ON.     Then  ^^^-  ^ 

6N^  =6A^  +  AN"^  =1+1  =  2. 

With  0  as  a  center  and  ON  as  radius,  strike  an  arc  inter- 
secting OX  at  B.     Then 

OW  =  2. 


18  THE   NUMBER   SYSTEM   OF   ALGEBRA        [Art.  15 

Thus  we  have  constructed  a  line-segment  on  OX  whose 
length  is  equal  to  V2  units,  if  we  use  the  customary  notation 
for  a  square  root.  But  we  can  easily  show  that  V2  is  not  a 
rational  number.     For,  if  it  Avere,  we  could  write 

(1)  v^  =  ^ 

where  p  and  q  are  positive  integers  without  a  common 
divisor,  since  we  may  think  of  the  rational  number  to  which 
V2  is  hypothetically  equal  as  being  expressed  in  its  lowest 
terms.  jSIoreover  q  cannot  be  equal  to  unity.  For  if  it 
were,  (1)  would  require  V2  to  be  an  integer,  and  clearly 
there  exists  no  integer  whose  square  is  equal  to  2.* 

The  prime  factors  (see  Art.  6)  of  p  are  all  different  from 
those  of  q  since,  by  hypothesis,  p  and  q  have  no  common 
divisor.  The  prime  factors  of  p^  are  the  same  as  those  of  p, 
each  of  them  occurring  twice  as  often  inp"^  as  in  p.  Similarly 
for  (f.  Consequently  every  one  of  the  prime  factors  of  p^ 
must  be  different  from  every  prime  factor  of  <f.  But  from 
(1),  if  true ^  we  can  conclude 

2=^  ov  p^=2q^ 

which  latter  equation  tells  us  that  p^  is  divisible  by  q^,  or  in 
other  words,  that  all  of  the  prime  factors  of  q^  are  also  prime 
factors  of  p^.  We  have  reached  a  contradiction,  thus  show- 
ing that  (1)  is  impossible. 

We  have  shown  that  the  line-segment  OB  (Fig.  8)  cannot 
be  represented  by  a  rational  number.  If  we  wish  to  speak 
of  the  measure  of  such  a  line-segment  as  a  number  at  all,  we 
must  therefore  introduce  a  new  kind  of  number.  These 
considerations  lead  us  to  make  the  following  postulate:  We 
agree  that  there  shall  correspond  to  every  Ihie-segment  on  OJT, 
a  positive  number,  ivhich  shall  serve  as  the  numerical  measure 
of  that  segment.  The  numbers  which  thus  correspond  to 
many  of  the  line-segments  of  OX  are,  of  course,  rational 

*  A  formal  proof  of  this  follows  immediately  from  the  monotonic  law  of 
multiplication. 


AuT.  15]     THE  EXISTENCE  OF  IRRATIONAL  NUMBERS     19 

numbers.  Irrational  positive  numbers  are  defined  as  the  nu- 
merical measures  of  those  line-segments  of  OX  which  are  not 
rational. 

Thus  y/2  is  an  irrational  number. 

We  can  now  say  that  to  every  line-segment  on  OX  there 
corresponds  a  positive  number  (rational  or  irrational);  and 
conversely,  to  every  positive  number  there  corresponds  a 
line-segment  on  OX. 

The  geometrical  constructions  which  have  been  indicated 
for  finding  the  sum,  difference,  product,  or  quotient  of  two 
rational  numbers  can  be  performed  whether  the  line-segments 
concerned  are  rational  or  not.  Consequently  these  construc- 
tions maybe  regarded  as  defining  the  sum,  difference,  product, 
or  quotient  of  any  two  positive  numbers  whether  they  are 
rational  or  not. 

In  formulating  these  definitions,  we  have  presupposed  certain  notions 
of  Geometry.  Since  every  definition  is  an  explanation  of  something  in 
terms  of  something  else,  not  all  things  are  definable.  The  most  funda- 
mental notions  of  a  mathematical  science  are  those  which  are  not  defined,  hut 
upon  which  all  of  the  later  definitions  are  based.  It  is  possible  to  define 
irrational  numbers  by  purely  arithmetical  considerations  (without  the 
use  of  Geometry),  but  those  definitions,  due  to  Dedekixi),^  Cantor,^ 
and  VV^EiERSTRASs,^  are  too  abstract  and  difficult  for  a  beginner. 

The  existence  of  irrational  numbers  is  seen  to  correspond  to  the 
existence  of  incommensurable  line-segments.  It  is  a  very  remarkable 
fact  that  this  was  known  to  the  Greeks  at  a  very  early  time.  The  dis- 
covery is  usually  ascribed  to  Pythagoras.^ 

EXERCISE  III 

1.  Prove  that  V8  is  irrational. 

2.  Prove  that  Vb  is  irrational. 

3.  Prove  that  VA:  is  irrational  if  k  is  a  positive  integer  which  is  not 
equal  to  the  square  of  another  integer. 

iR.  Dedekind  (1831-1916).  German. 
2  G.  Cantor,  born  184.5  in  Potroiirad. 
8  K.  Weierstrass  (1H15-1H!>7),  German. 

^  Pythagor.as  and  his  puiuls,  tlio  Pythagoreans,  flourished  five  or  .six  centuries 
before  Christ  in  various  countries  then  under  Greek  influence. 


20  THE   NUMBER   SYSTEM   OF   ALGEBRA        [Art.  16 

4.  Review  from  your  Geometry  the  method  of  constructing  a  line- 
segment  of  lengtli  ^ ah  if  the  line-segments  of  length  a  and  h  are  given. 
Apply  this  method  to  the  construction  of  V^,  Vo,  yJk. 

5.  Choose  a  unit  of  length.  Construct  V3  and  V5,  V3  +  \/5, 
V5-  V3,   V3  X  V5,   V.5  ^  V.5,   V5  -  V3. 

6.  Observe  that  the  constructions  for  a  +  />,  a  —  i,  «ft,  a/6,  and  Va  can 
all  be  performed  by  using  ruler  and  compass.  Show  that  if  a,  h,  c,  d,  etc. 
are  given  line-segments,  it  is  possible  to  find  a  ruler  and  compass  con- 
struction for  any  line-segment  which  can  be  obtained  from  a,  h,  c,  d,  etc. 
by  a  finite  number  of  additions,  subtractions,  multiplications,  divisions, 
and  extractions  of  square  roots. 

16.  Negative  numbers  and  zero.  If  we  prolong  the  line  OX 
backward  from  0  in  the  direction  OX'  (see  Fig.  9),  we  may 

lay  off  line-segments,  such  as  OC 

x'    <,  I,   I     '  "^1  \  "^' X    and  OB' ,  in  either  of  two  opposite 

p,j^  q  directions.    These  segments  differ, 

not  only  in  length,  but  also  in 
direction.  We  agree  to  indicate  this  difference  hi  direction  hy 
the  use  of  the  signs  +  ayid  — . 

Thus  we  say  that  the  number  which  corresponds  to  OC  is  -|-  3  and 
that  which  corresponds  to  OB'  is  —  2. 

The  new  numbers  (such  as  —  2)  which  are  introduced  in 
this  way  are  called  negative  numbers.  If  we  agree  further 
that  the  line-segment  00,  of  no  length,  which  joins  0  to  itself, 
shall  be  represented  by  the  symbol  or  number  0  (zero), 
every  line-segment  of  X' X  which  has  0  as  origin  ivill  determine 
a  number,  positive^  zero,  or  negative,  according  as  the  terminus 
of  this  line-segment  is  to  the  right  of  0,  coincides  with  0,  or  is  to 
the  left  of  0. 

Every  point  P  of  XX'  may  be  thought  of  as  the  terminus 
of  a  line-segment  OP  which  has  0  as  its  origin.  If  we  label 
the  point  P  with  the  positive  or  negative  number  whicli  cor- 
i-esponds  to  tlie  segment  OP,  we  obtain  a  scale  of  positive  and 
negative  numbers,  as  in  Fig.  9.  A  familiar  instance  of  such 
a  scale  is  furnished  by  an  ordinary  thermometer.  It  makes  no 
essential  difference  on  which  side  of  the  scale  we  mark  the 


Art.  17]  DIRECTED   LINES  21 

positive  numbers,  provided  we  use  the  opposite  side  for  the 
negative  numbers. 

We  have  iutroduced  positive  and  negative  numbers  to  indicate  opposi- 
tion in  direction,  such  opposition  as  is  expressed  in  ordinary  language  by 
the  terms  right  and  left,  above  and  below,  north  and  south,  east  and  icest. 
But  there  are  many  other  instances  where  a  similar  kind  of  opposition  is 
to  be  indicated,  although  not  of  geometric  character,  and  where  the  use 
of  negative  numbers  is  of  great  importance.  A  few  examples  of  this  are 
indicated  by  the  terms,  before  and  after,  temperatures  above  and  below  zero, 
profit  and  loss,  credit  and  debit.  If  one  of  the  members  of  any  of  these 
pairs  be  represented  by  a  positive  number,  the  other  member  may  be 
represented  by  a  negative  number. 

17.  Directed  lines  and  directed  line-segments.  When  a 
line  XX'  has  been  provided  with  a  scale  of  positive  and  nega- 
tive numbers,  as  in  Art.  16,  we  shall  henceforth  speak  of  it 
as  a  directed  line,  its  positive  sense  being  that  one  which  is 
directed  from  the  origin  toward  the  side  which  represents 
the  positive  numbers.  In  our  figures  we  shall,  from  now 
on,  indicate  the  positive  sense  of  a  directed 
line  by  placing  a  +  sign  near  the  end  of  that 
portion  of  the  line  which  actually  appears  in 
the  figure.     (See  Fig.  10.) 

A  line-segment  is  a  finite  portion  of  a  line  and  may  be 
described  by  naming  its  end-points,  such  as  AB  in  Fig.  10. 
But  if  we  think  of  it  as  a  directed  line-segment  we  must  dis- 
tinguish between  AB  and  BA.  We  may  think  of  a  directed 
line-segment  as  being  generated  by  the  motion  of  a  poinU  which 
moves  from  one  of  its  end-points  toward  the  other  without  ever 
changing  the  direction  of  its  motion.  That  end  of  the  directed 
line-segment  from  which  the  generating  point  starts  its 
motion  is  called  the  origin  of  the  directed  line-segment. 
The  other  end-point  is  called  its  terminus.  In  speaking  of 
a  directed  line-segment  we  shall  name  its  origin  first.  Thus, 
in  Fig.  10,  AB  is  the  directed  line-segment  which  has  A  as 
origin  and  B  as  terminus.  The  directed  line-segment  BA 
has  the  same  length  as  AB,  l)ut  the  opposite  direction. 

If  a  directed  line-segmeiil  lies  on  a  directed  line,  its  direc- 
tion or  sense  may  be  the  same  as  that  of  the  directed  line  or 


22  THE   NUMBER   SYSTEM   OF    ALGEBRA        [Art.  17 

else  opposite.  A  directed  line-segment,  which  lies  on  a  di- 
rected line,  shall  be  regarded  as  having  a  positive  or  negative 
number  as  its  measure  according  as  it  has  the  same  direction  as 
the  directed  line  or  the  opposite  direction. 

Thus  ill  Fig.  10,  if  the  length  of  AB  is  5  units,  we  may  write  AB  =  5 
and  BA  —  —  5.  The  measure  of  vIjB  is  5,  that  of  BA  is  —  5.  We  shall 
use  the  same  symbol  ^.6  to  represent  both  the  line-segment  itself  and 
its  measure. 

The  absolute  value  of  a  line-seg7nent,  or  its  numerical  value,  is 
always  a  positive  number  expressing  how  many  units  there  are 
in  its  leyigth.  We  use  the  notation  \  AB  \,  placing  AB  between 
two  vertical  lines,  to  represent  the  absolute  value  of  AB. 

Thus  in  Fig.  10,  \AB\  =  5,  and  \BA\^  5.  The  first  of  these  state- 
ments is  read :  the  absolute  value  oi  AB  is  equal  to  5. 

A  directed  line-segment  on  a  directed  line  is  said  to  be  in  its 
standard  position  if  the  origin  of  the  line-segment  coincides  with 

the  origin  of  the  scale. 

In  Fig.  11,  the  directed  line-segment 
AB,  whose  measure  is  +  3,  is  not  in  its 
standard  position.  If  it  be  placed  in 
standard  position  it  will   coincide  with 

OC.     Its  measure  will  still  be   +  3.     This  remark  also  illustrates  the 

following  statement. 

Two  directed  line-segments  on  the  same  directed  line  have  the 
same  measure  if  their  lengths  and  senses  are  the  same.  Two 
such  segments  will  henceforth  be  regarded  as  equal  whether 
their  origins  coincide  or  not. 

From  what  has  been  said  it  is  clear  that,  if  A  and  B  are 
two  points  on  a  directed  line,  the  measures  of  the  directed 
line-segments  AB  and  BA  will  be  numerically  equal,  but 
opposite  in  sign,  that  is, 

(1)  AB=-  BA. 

The  following  theorem  is  also  important.  If  A,  B,  C  are 
any  three  points  on  a  directed  line,  then 

(2)  AB  +  BC=AC, 


Art.  17] 


DIRECTED    LINES 


23 


where  AB^  B  (7,  and  A  0  are  the  measures  of  the  corresponding 
directed  line-segments. 

Proof.  The  symbol  +  in  equation  (2)  is  used  to  indicate 
addition  as  defined  geometrically  on  page  12,  and  again  on 
page  24.  According  to  this  definition,  to  obtain  the  sum  of 
two  directed  line-segments  we  place  the  origin  of  the  second 
segment  on  the  terminus  of  the  first,  and  then  join  the  origin 
of  the  first  segment  to  the  terminus  of  the  second.  If  we 
apply  this  definition  to  the  sum  AB  +  BO  we  obtain  AC,  no 
matter  in  wliat  order  the  three  points  A,  B,  and  O  are  ar- 
ranged on  the  line.  Thus  the  theorem  is 
proved. 

Figure  12  shows  three  of  the  six  possible  arrange- 
ments of  the  three  points,  namely  those  three  in 
which  the  measure  of  AC  is  positive.  Fig.  12  (a), 
in  which  AB  and  BC  also  have  positive  measures, 
teaches  us  nothing  new.  In  Fig.  12  (i),  AB  and 
CB  are  positive,  but  BC  is  negative.  According 
to  our  theorem,  which  is  true  in  all  cases,  we  have 
AB  +BC  ^AC.  But  Fig.  12  (i)  shows  that 
AB  —  CB  =  AC.  Therefore  we  obtain  the  same 
result  whether  we  add  to  AB  the  negative  segment 
BC  or  subtract  from  AB  the  positive  segment  CB. 
suit  from  Fig.  12  (c). 

The  following  is  a  simple  corollary  of  the  above  theorem. 

J^  A,  B,  (7,  JD  are  any  four  points  of  a  directed  line.,  tve  have 

the  relation 

(3)  AB+BC+  OB  =  AI) 

between  the   measures  of  the  directed  line-segments  AB.,  BC, 
CD,  and  AD. 

In  fact,  by  the  theorem  just  proved,  we  have 

AB  +  BC=AC    AC+  CD  =  AD, 

so  that  we  find  by  addition 

AB  +  BC-{-AC+CD  =  AC+AD 

which  reduces  to  (3)  if  we  subtract  AC  from  both  members. 


Similar  remarks  re- 


24  THE   NUMBER   SYSTEM   OF   ALGEBRA        [Art.  18 

It  may  now  be  proved  by  mathematical  induction  (see 
Art.  263)  that,  if  A,  B,  C,  .  .  .  M,  N  are  any  finite  7iumber 
of  points  on  a  directed  line,  then 

(4)  AB  +  BC-{--  -\-MN=AK 

18.  Addition  and  subtraction  of  positive  and  negative  num- 
bers. The  fundamental  operations  of  arithmetic  have  been 
defined,  so  far,  only  for  positive  numbers.  These  definitions 
may,  in  part,  be  applied  without  essential  change  to  the  case 
where  some  or  all  of  the  numbers  involved  are  negative. 
But  since  nothing  was  said,  at  the  time,  concerning  such 
cases,  it  becomes  necessary  to  re-define  these  operations  so 
as  to  take  into  account  all  of  the  possibilities. 

We  define  addition  geometrically,  as  follows  : 

To  add  a  number  a'  to  a  number  a,  we  first  construct  two 
directed  line-sec/ments  OA  and  O'A'  on  a  directed  line  I,  such 
that  the  measures  of  OA  and  O'A'  are  equal  to  the  numbers  a 
and  a'  respectively.  We  then  place  the  origin  0'  of  O'A'  upon 
the  terminus  A  of  OA.  The  directed  line-segment  OA',  which 
joins  the  origin  of  the  first  segment  to  the  terminus  of  the 
second,  will  have  a  -{-  a'  as  its  measure. 

In  each  of  the  cases  represented  in  Fig.  13,  OA'  represents  the  sum  of 

OA  and  O'A'.     Jn  all  three  cases  OA  is  positive.     In  the  first  case  O'A' 

is  also  positive.     In  the  second  case  O'A'  is  nega- 

+1     tive  but  of  absolute  value  less  than  OA,  so  that  the 

+1     sum  OA'  is  positive.     In   the  third  case  O'A'  is 

+i     negative  and  numerically  greater  than  OA,  so  that 

the  sum  OA'  =  OA  +-  O'A'  is  negative. 

This  definition  enables  us  to  prove  easily  that  addition  of 
numbers  (positive  or  negative)  still  satisfies  the  first  three 
fundamental  laws  of  Art.  2,  that  is,  addition  gives  a  unique 
result,  it  is  commutative  and  associative.  The  question 
whether  addition  is  also  monotonie  will  be  discussed  a  little 
later. 

If  the  sum  s  of  two  numbers  and  one  of  the  numbers  a 
are  given,  the  problem  to  find  the  other,  a',  may  be  expressed 


o 

A 

a' 

o' 

a' 

o 

A 

0 

Fig. 

A 

13 

Art.  18]  ADDITION   AND   SUBTRACTION  25 

as  follows:  wliat  number  added  to  a  will  give  the  sum  «? 
We  usually  write 

(1)  a'  =  s—a 

and  speak  of  the  process  of  finding  a'  as  subtraction.  «  is 
called  the  minuend,  a  the  subtrahend,  and  a'  the  difference. 
As  long  as  we  dealt  with  positive  numbers  only,  subtraction 
was  impossible  when  the  subtrahend  was  greater  than  the 
minuend.  The  introduction  of  negative  numbers  frees  us 
from  this  restriction. 

Thus  in  Fig.  13,  let  the  directed  line-segment  OA'  repre- 
sent the  minuend,  and  OA  the  subtrahend.  Then,  according 
to  (3)  of  Art.  17,  we  have  in  all  cases 

OA  +  AA'  =  OA'  or   OA  +  O'A'  =  OA' 

so  that  we  must  add  O'A'  to  OA  (the  subtrahend)  in  order 
to  obtain  OA'  (the  minuend)  as  a  sura.  Therefore  O'A'  is 
the  required  difference,  that  is, 

(2)  O'A'  =  OA'  -  OA. 

After  a  slight  change  of  notation  we  may  formulate  this  re- 
sult in  the  form  of  a  geometric  rule  for  subtraction. 

To  subtract  a  number  a'  (subtrahend^  from  a  number  a 
(minuend)  we  first  construct  two  directed  line-segments  OA  and 
O'A'  on  a  directed  line  I,  such  that  the  measures  of  OA  and 
O'A'  are  equal  to  a  and  a'  respectively .  We  then  place  the 
termini  of  these  two  segments  so  that  they  shall  coincide.  The 
directed  line-segment  which  theri  joins  the  origin  of  the  minuend 
to  the  origi^i  of  the  subtrahend  will  represent  the  difference  iii 
magnitude  and  sign. 

These  constructions  easily  lead  to  the  following  familiar 
remarks. 

27ig  addition  of  a  negative  number  is  equivalent  to  the  sub- 
traction of  a  positive  number  of  the  same  absolute  value.,  that  is., 

(1)  a  +  (-5)=a-6. 


26  THE   NUMBER  SYSTEM  OF   ALGEBRA        [Art.  19 

The  subtraction  of  a  negative  number  is  equivalent  to  the 
addition  of  a  positive  number  of  the  same  absolute  value,  that  is, 

(2)  a-(-5)=a+5. 

Since  subtraction  of  any  number  may  tlierefore  always  be 
regarded  as  addition  of  a  number  of  the  same  absolute  value 
but  of  opposite  sign,  we  may  from  now  on  suppress  any  ex- 
plicit mention  of  the  laws  of  subtraction.  They  are  included 
in  the  laws  of  addition. 

The  following  statements  are  also  immediate  consequences 
of  our  geometric  definitions  : 

(3)  a -\- 0  =  a,       a  —  0  =  a,       a  —  a  =  0,       «  +  (— a)=0. 

19.   Multiplication  of  positive  and  negative  numbers.     We 

have,  so  far,  defined  multiplication  only  for  the  case  where 
both  factors  are  positive.  In  our  attempt  to  formulate  an 
appropriate  definition  for  multiplication  for  the  case  where 
one  or  both  factors  are  negative,  we  return  for  a  moment  to 
the  simplest  case  of  all  when  both  factors  are  positive  in- 
tegers. In  that  case  multiplication  reduces  to  a  repeated 
addition.     Thus,  for  instance, 

(1)  .         5  X  3  =  5  -h  5  -f-  5  =  15. 

If  the  multiplicand  is  negative,  equal  to  —  5  for  instance, 
while  the  multiplier  is  a  positive  integer,  we  naturally  ex- 
tend this  by  saying  that  —  5  multiplied  by  3  shall  mean 
—  5  —  5  —  5,  that  is, 

(2)  (_  5)  .  3  =  -  5  -  5  -  5  =  -  15. 

If  the  multiplier  is  negative,  it  becomes  impossible  to  think 
of  multiplication  as  repeated  addition.  But  we  know  that 
if  both  factors  are  positive,  multiplication  is  commutative 
so  that  5  •  3  =  3  •  5.  If  multiplication  is  to  be  defined  in 
suph  a  way  as  to  remain  commutative,  even  if  one  factor 
is  negative,  we  must  liave 

(3)  3.(-5)  =  (-5).3 


Aht.  19]  MULTIPLICATION  OF   NUMBERS  27 

and  therefore,  on  account  of  (2) 

(4)  3.(-5)  =  -15. 

Let  us  now  think  of  these  numbers  as  the  measures  of 
directed  line-segments.  In  cases  (1)  and  (2)  the  multiplier 
is  positive  (equal  to  +  3),  and  the  ec^uations  show  that  the 
multiplication  by  the  positive  multiplier  8  merely  stretches 
the  line-segment  which  represents  the  multiplicand  in  the 
ratio  of  3 :  1  without  altering  its  direction.  For  a  positive 
multiplier,  the  product  has  the  same  sign  as  the  multiplicand. 
But  (4)  shows  that  multiplication  by  —  5  not  merely 
stretches  the  line-segment  in  the  ratio  5 :  1  but  also  reverses 
its  direction.  These  remarks  suggest  the  following  defini- 
tion of  multiplication  for  positive  or  negative  numbers  : 

To  multiply  a  directed  line-segment  a,  of  a  directed  line,  by 
a  positive  line-segment  b,  we  merely  stretch  a  in  the  ratio  of 
5:1.  To  mtdtiply  a  by  a  negative  line-segment  6,  we  stretch 
a  in  the  ratio  of  \b\'.  1  and  also  turn  the  resulting  line- 
segment  around  through  tivo  right  angles,  thus  reversing  its 
direction. 

We  may  re-formulate  the  substance  of  this  definition  with- 
out any  reference  to  Geometry  as  follows : 

T/ie  numerical  value  of  a  product  of  two  factors  is  equal  to 
the  product  of  the  numerical  values  of  the  factors.  The  sign  of 
the  product  is  plus  if  both  factors  have  the  same  sign,  and  minus 
if  the  signs  of  the  ttvo  factors  are  opposite. 

This  statement  includes  the  familiar  rules  expressed  by 
the  symbolic  equations 

(5)  (  +  )(+)= -f,    (-)(-)=+,    (  +  )(_)  =  _, 

(-)(  +  )  =  -. 

It  should  be  noted  that  these  rules  have  not  been  proved. 
They  are  a  part  of  the  definition  of  multiplication. 

Our  definition  of  multiplication  is  completed  by  consider- 
ing the  case  where  one  factor  is  equal  to  zero. 


28  THE   NUMBER   SYSTEM   OF   ALGEBRA        [Art.  20 

If  one  of  the  factors  of  a  product  is  equal  to  zero,  the  product 
is  also  equal  to  zero,  that  is, 

(6)  a  X  0  =  0  X  a  =  0. 

If  a  product  is  equal  to  zero,  at  least  one  of  its  factors-  must 
vanish. 

For  our  definition  of  multiplication  gives  us  a  non-vanishing  product 
whenever  both  factors  are  different  from  zero.  Consequently  both  factors 
cannot  be  different  from  zero  if  the  product  is  zero. 

The  geometric  construction  for  multiplication  given  in 
Art.  12  becomes  applicable  also  to  the  case  where  one  or 
both  factors  may  be  negative,  if  we  think  of  the  lines  OX  and 
Oy  as  directed  lines  and  lay  off  the  line-segments  which 
represent  the  factors  in  one  direction  or  another  upon  OX 
and  OY  according  to  their  signs.  It  will  usually  be  most 
convenient  for  this  purpose  to  draw  OX  and  OY  at  right 
angles  to  each  other. 

20.   Division  by  a  positive  or  negative  divisor.     As  in  the 

case  of  positive  numbers,  we  think  of  division  as  the  opera- 
tion inverse  to  multiplication.  That  is,  if  D  is  the  dividend, 
d  the  divisor,  and  q  the  quotient,  the  problem  involved  in 

the  statement  ._,       , 

V  ^  d  =  q 

is  that  of  finding  a  number  q  such  that 

d     q  =  D, 

where  d  and  D  are  given.  We  have  seen  how  to  solve  this 
problem,  both  arithmetically  and  geometrically,  if  D  and  d 
are  both  positive.  From  what  was  said  in  Art.  19  about 
multiplication  it  appears  immediately  that  the  following  rule 
will  hold : 

To  divide  D  hy  d  where  either  or  both  numbers  may  be  nega- 
tive, first  proceed  as  though  both  were  positive.  Then  give  the 
quotient  the  plus  or  minus  sign  according  as  D  and  d  have  the 
same  sign  or  opposite  signs. 


Arts.  21,22]  THE   MONOTONIC   LAWS  29 

21.  Division  by  zero.  To  divide  I)  hy  d  means  to  find  a 
number  q  such  that 

(1)  D  =  dq. 

If  D  is  not  zero  and  t?  =  0,  this  equation  involves  a  contra- 
diction. For,  if  rf  =  0,  the  right  member  of  (1)  will  have 
the  value  zero,  no  matter  what  may  be  the  value  of  q^  and 
this  contradicts  the  assumption  that  D  is  not  zero.  Con- 
sequently, the  notion  of  a  quotient  formed  from  a  non-vanish- 
ing dividend  and  a  zero  divisor  is  self -contradictor  i/ , 

This  contradiction  disappears  if  dividend  and  divisor  are 
both  equal  to  zero.  In  this  case,  however,  equation  (1) 
will  be  satisfied  by  any  number  q  whatever,  so  that  the  quo- 
tient is  entirely  undetermined  if  both  D  and  d  are  equal  to 
zero.  Thus,  the  notion  of  a  quotient  formed  from  a  zero  divi- 
dend and  a  zero  divisor,  while  not  self -contradictory^  is  useless 
since  it  gives  no  determinate  result. 

In  neither  case  can  we  make  use  of  division  by  zero.  For 
these  reasons,  division  by  zero  must  be  rigorously  excluded 
from  algebra.  Whenever,  in  any  algebraic  argument,  we 
perform  a  division  we  must  always  either  prove  that  the 
divisor  is  different  from  zero,  or  else  we  must  state  explicitly 
that  our  conclusion  has  been  proved  only  for  those  cases  in 
which  the  divisor  is  not  equal  to  zero.  Neglect  of  these 
precautions  easily  leads  to  the  most  absurd  results. 

22.  The  monotonia  laws  for  positive  and  negative  numbers. 

The  monotouic  laws  for  positive  numbers  are  concerned  with 
inequalities  and  contain  the  symbols  <  and  >.  Before  we 
can  speak  about  the  monotonic  laws  for  negative  numbers, 
we  must  extend  the  significance  of  these  symbols  so  as  to 
make  them  applicable  to  negative  as  well  as  positive  num- 
bers. In  order  to  do  this,  let  us  think  of  the  scale  of  positive 
and  negative  numbers,  and  let  tis  think  of  a  point  ivhich  moves 
along  this  scale  in  the  direction  from  the  negative  toward  the 
positive  numbers.  We  shall  then  sag  that  a  <.  b  {a  precedes  b, 
or  a  is  less  than  6)  if  this  moving  point  reaches  a  before  it 
reaches  b. 


30  THE   NUMBER   SYSTEM  OF   ALGEBRA        [Art.  22 

Moreover,  if  a  <ih  we  say  also  that  h  >  a,  that  is,  h  follows  a,  or  h  is 
greater  than  a. 

In  accordance  with  tliis  definition  we  have,  for  instance,  3  <  5,  —  2  <  0, 
—  5  <  —  3,  although  5  is  greater  than  3. 

It  is  now  easy  to  see  that,  if  we  use  the  signs  of  inequality 
in  accordance  with  this  definition,  addition  will  still  be 
monotonia  even  though  one  or  all  of  the  numbers  concerned 
may  be  negative.  That  is,  if  a>  b,  then  a  -\-  c  >  b  +  c,  whether 
a,  5,  c  are  positive  or  negative. 

The  monotonic  law  for  multiplication.^  however.,  undergoes  a 
slight  but  essential  modification.  If  a  >  b,  theti  ac  >  be  only  if 
c  is  positive.  If  c  is  negative  the  conclusion  to  be  drawn  from 
a>  b  is,  ac  <  be. 

Thus,  from  —  4  <  —  3  follows  —  8  <  —  6  if  yve  multiply  both  mem- 
bers of  this  inequality  by  +  2.  But  if  we  multiply  by  —  2  we  must 
change  the  sense  of  the  inequality  sign,  since  +  8  >  +  6. 

The  monotonic  laws  are  essential  in  all  questions  involv- 
ing inequalities. 

Except  for  this  slight  change  in  the  monotonic  law  for  imdti- 
plication,  the  nine  fundamental  laws  of  Art.  2  hold  for  all  posi- 
tive and  negative  ^lumbers. 

We  have  already  proved  this  for  the  first  three  laws  in 
Art.  18.  The  general  validity  of  law  IV  has  just  been 
proved.  The  truth  of  laws  V,  VI,  VII,  and  IX  for  the  case 
of  positive  and  negative  numbers  follows  at  once  from  our 
definition  of  multiplication,  and  we  have  just  seen  how  the 
eighth  law  must  be  modified  so  as  to  remain  valid  in  the 
case  of  a  negative  multiplier. 

EXERCISE  IV 

Perform  the  following  indicated  operations  arithmetically  and  also 
graphically. 

1.  5 +  (-3).  4.    6- (-8).  7.    (-8) -(-2). 

2.  5 -(-3).  5.    (_7)(-4).  8.    [5  +  (- 3)][- 7  +  9]. 

3.  (-6) -7.  6.-8-4.  9.    [5  +  (- 3)]  -^  [- 7+ 9]. 


Art.  23]     DIRECTED   LINE-SEGMENTS   IN   A   PLANE  31 

Simplify  the  following  expressions.  Justify  each  step  of  the  trans- 
formation by  appealing  to  one  of  the  fundamental  laws. 

10.  (a  +  b)(c-d).  "•     iriy^-'^' 

11.  (a-h)(c-(0-  14.    (£_lj(^  +  iy 

16.  Prove  that  |  «  +  ft  |  =  |  a  |  +  |  6|  if  «  and  h  have  the  same  sign,  but 

that  |a-f-6|<|n|+|ft|if  the  signs  of  a  and  h  are  not  alike.     Consequently 

we  shall  always  have  ,         ,  ,  ^       , 

•^  \'-i  +  fj\<[a\  +  \b\, 

where  the  symbol  <  is  read  is  not  greater  than. 

17.  Show  that  a-  +  h- >2ah\i  a  and  h  are  not  equal. 

Solution.  If  a  and  b  are  not  equal,  a  —  b  may  be  positive  or  negative, 
but  {a  -hy=  {a-b)(a  -  b)  will  be  positive.     (See  Art.  19.)     There- 

^°''®  «2  -Oah  +  b^  >  0. 

According  to  the  monotonic  law  of  addition,  we  may  add  '2  ab  to  both 
members  without  changing  the  sense  of  the  "inequality.     Therefore 

a-  +  b-  >  2  ab 
as  was  to  be  proved. 

18.  Show  that  a-  +  b-  +  c'->ab  +  (ic  +  be. 

19.  Show  that  >  -^^-^  if  a  and  b  are  positive. 

2  a  +  ft 

20.  Show  that  (/  +  -  >  2  if  '/  is  a  positive  number  not  equal  to  either 

a 

1  or  0.     Why  this  last  restriction  ? 

21.  Point  out  the  error  in  the  following  argument.  Let  x  =  a.  Then 
we  find  successively  x^  =  ax,  x-  —  a^  =  ax  —  a-,  (x  —  (i)(x  +  a)  — 
a(x  —  fl),  X  +  rt  =  a.     But  since  x  =  a,  this  gives  2  a  —  a  and  therefore 

2  =  L 

23.  Directed  line-segments  in  a  plane.  We  have  repre- 
sented positive  and  negative  numbers  by  directed  line-seg- 
ments on  a  directed  line,  which  we  shall  henceforth  call  the 
r-axis.  From  the  point  of  view  of  Plane  Geometry,  however, 
the  directed  line-segments  of  this  particular  line,  the  ^--axis, 
are  of  no  greater  importance  than  the  directed  line-segments 


32  THE   NUMBER   SYSTEM   OF   ALGEBRA        [Art.  23 

of  any  other  line.  We  propose  therefore  to  generalize  the 
notion  of  a  directed  line-segment,  by  admitting  that  such  a 
segment  may  be  situated  anywhere  in  the  plane,  not  neces- 
sarily upon  the  a;-axis. 

We  shall  regard  tivo  directed  line-segments  (such  as  AB  and 
A' B'  in   Fig.  14)  as  equal,  if  they  have  the  same  length,  if 
they  are  on  the  same  or  parallel  lines, 
and  if  they  have  the  same  sense. 

In   applying   this    definition    of    equality, 

*-*  we   must   take  into  account  all  of  the  three 

characters  mentioned.     Thus,  in  Fig.  14,  AB 

and  B'A  '  are  not  equal  although  they  have  the  same  length,  and  are  on 

parallel  lines.     Their  senses  are  opposite. 

As  in  Art.  17  we  may  think  of  the  directed  line-segment 
as  generated  by  a  point  which  starts  from  one  end-point  and 
moves  without  reversing  its  direction,  until  it  reaches  the 
other  end-point.  Thus,  we  reach  the  notion  of  the  origin  and 
terminus  of  such  a  directed  line-segment.  In  Fig.  14  the 
origin  of  AB  is  A,  its  terminus  is  B. 

.  We  may  always  construct  a  directed  line-segment  OP 
with  its  origin  at  the  origin  of  the  number-scale  of  the  a^-axis, 
and  equal  to  any  given  directed  line-segment  AB.  (See 
Fig.  14.)  We  then  say  that  the  line-segment  AB  has  been 
placed  in  its  standard  position  OP.  The  length  r  of  the 
line-segment  is  called  its  modulus  or  absolute  value  and  is 
always  a  positive  number  ;  the  angle  6  which  it  makes  with 
the  2;-axis  is  called  its  amplitude  or  argument.  We  see  that 
two  numbers  (r  and  ^)  must  be  given  before  we  can  regard 
the  directed  line-segment  OP,  or  its  equal 
AB,  as  known. 

There  is  a  second  way  of  describing  such 
a  directed  line-segment.  We  introduce  a 
second  scale  of  numbers,  just  like  the  a:-scale, 
on  a  line  perpendicular  to  the  rc-axis  through 
the  point  0.  We  place  the  origin  of  this  second  scale  (the 
y-scale)  also  at  0  and  place  its  positive  end,  as  in  Fig.  15, 
so  as  to  be  above  the  a;-axis.     If  now  we  project  the  directed 


+y 

.v 
y 

.r 

F 

^ 

y 

0 

•c     J 

I 

i+y 

R 

F 

/ 

E 

-V 

A\      > 

N 

-yP 

0 

M 

C    D 

Art.  23]     DIRECTED   LINE-SEGMENTS   IN   A   PLANE  33 

line-segment  OP  on  the  aj-axis  and  also  on  the  y-axis,  we 
obtain  two  directed  line-segments,  OM  and  OiV,  to  each  of 
which  will  correspond  as  its  measure  a  positive  or  negative 
number,  on  its  own  scale.  We  shall  call  these  numbers  x 
and  t/  respectively.  Their  numerical  values  give  the  lengths 
of  the  segments  OM  and  OiV  respectively,  while  their  signs 
indicate  whether  M  is  to  the  right  or  left  of  0,  and  whether 
iVis  above  or  below  0.  (In  Fig.  15,  x  and  y  are  both  posi- 
tive.) The  numbers  x  and  y  are  called  the  components  of 
the  directed  line-segment  OP.  It  is  clear  that  any  directed 
line-segment  AB,  which  is  equal  to  OP  but 
which  is  not  in  its  standard  position,  will 
have  the  same  components  as  OP.  See  Fig. 
16,  where  OP  and  AB  are  equal  directed 
line-segments,  and  where  the  components 
CD  and  UF  of  AB  are  equal  to  the  com- 
ponents OM  and  OiV  of  OP  respectively. 

If  both  of  the  components  of  OP  are  given,  in  magnitude 
and  sign,  we  can  clearly  find  OP  itself  by  a  simple  construc- 
tion. Therefore  the  components  of  a  directed  line-segment 
determine  it  just  as  completely  as  its  modulus  and  amplitude. 

If  we  are  acquainted  with  Trigonometry  we  can  easily 
compute  the  components  when  the  modulus  and  amplitude 
are  given.*     From  Fig.  15  we  see  at  once  that 

(1)  X  =  r  cos  6,  y  =  r  sin  6 

We  have  agreed  already  that  the  modulus  r  shall  always 
be  regarded  as  a  positive  number.  It  is  desirable  to  define 
the  amplitude  a  little  more  precisely  than  we  have  done  so 
far.  The  amplitude  0,  of  a  directed  line-segment  OP  in  its 
standard  position,  is  the  angle  generated  hy  a  line  which  origi- 
nally coincides  with  the  positive  x-axis  and  tvhich  rotates  around 
0  as  a  center.,  in  a  counter-clocku'ise  direction  (toivard  the  posi- 
tive y- axis').,  until  it  coincides  with  OP. 

*  Students  who  have  not  studied  Trigonometry  may  omit  those  parts  of  this 
chapter  iu  whicrh  Triijonometry  is  used.  Alternative  treatments  are  given  not 
involving  a  knowledge  of  Trigonometry. 


34  THE   NUMBER   SYSTEM   OF   ALGEBRA        [Art.  24 

Therefore  the  amplitude  need  not  be  an  acute  angle.     Thus,  in  Fig. 
17  the  amplitude  of  OP  is  obtuse,  that  of  OP'  is  in  the  third,  and  that 
of  OP"  in  the  fourth  quadrant. 

For  some  purposes  it  is  important  to 
remember  that  the  detinition  of  ampli- 
tude just  given,  does  not  determine  the 
amplitude  without  ambiguity.     The  last 

„  words  of  the  definition  do  not  read  :   until 

Fig.  17 

it  coincides  with  OP  for  the  first  time. 
Consequently  we  may  add  any  integral  multiple  of  360°  to  the 
amplitude  of  a  directed  line-segment,  and  the  resulting  angle 
may  still  be  regarded  as  the  amplitude  of  that  line-segment. 
Obviously  we  may  now  look  upon  the  negative  numbers 
of  the  a;-axis  as  corresponding  to  directed  line-segments  of 
amplitude  180°  or  tt  radians. 

Thus  the  modulus  of  —  4  is  4;  its  amplitude  is  180°. 

The  truth  of  equations  (1)  for  the  case  where  6  is  an  acute 
angle  is  evident  from  Fig.  15.  But  these  equations  are  also 
true  if  0  is  in  any  quadrant.  That  this  is  so  will  become 
apparent  if  we  compare  these  equations  with  those  which 
are  used  for  the  purpose  of  defining  the  sine  and  cosine  of  a 
general  angle. 

Either  from  (1)  or  directly  from  Fig.  15,  we  find 

(2)  r  =  +  V.r2  +  ^2,  tan  9  =  •^, 

thus  enabling  us  to  compute  the  modulus  and  amplitude  of 
a  directed  line-segment  when  its  components  are  given. 
However,  the  second  equation  alone  does  not  suffice  to  deter- 
mine 6  unambiguously  as  to  its  quadrant,  since  there  are 
two  angles  in  the  first  four  quadrants  (differing  by  180°) 
which  have  the  same  tangent.  However,  since  x  and  y  are 
given,  and  r  is  positive,  equations  (1)  tell  us  by  inspection 
the  sign  of  sin  6  and  cos  ^,  and  therefore  the  quadrant  of  6. 

24.  The  complex  numbers.  We  have  seen  that  a  directed 
line-segment,  such  as  OF  in  Fig.  15,  is  determined  in  mag- 
nitude and  direction  by  jneans  of  its  components,  x  and  y. 


Art.  24] 


THE   COMPLEX  NUMBERS 


35 


•2    -1      o 


+y 

+3t 


In  particular,  if  ?/  =  0,  OP  becomes  a  directed  line-segment 
on  the  rr-axis  and  may  be  regarded  as  representing  a  positive 
or  negative  number,  as  in  Art.  16.  If  a:  =  0,  OP  becomes 
a  directed  line-segment  on  the  y-axis,  and  we  might  again 
associate  with  such  a  line-segment  a  positive  or  negative 
number.  Unless  we  adopt  some  notation,  therefore,  which 
will  enable  us  to  distinguish  at  a  glance  between  directed 
line-segments  on  the  a:-axis  and  directed  line-segments  on 
the  y-axis,  the  number  +  3  might 
be  thought  of  as  a  line-segment 
on  either  axis.  In  order  to  avoid 
this  ambiguity,  we  label  the  scale 
of  numbers  on  the  ?/-axis  with  tlie 
symbols  +  i,  +2  i,  +  3  z,  etc., 
—  i,  —  2  z,  —  3  i,  etc.  (See  Fig. 
18.)  The  four  directed  line-seg- 
ments OA,  OB,  00,  OD  in  Fig. 
18,  all  of  length  2  but  having  different  directions,  may 
now  very  briefly  be  denoted  by  -h  2,  -f  2  2,  —  2,  —  2  i 
respectively. 

More  generally  ive  may  noiv  represent  any  directed  line-seg- 
ment OP,  ivhose  componeyits  are  x  and  y,  hy  the  symbol 

z  =  X  +  yi. 
Thus  in  Fig.  18,  OP  will  be  represented  by  3  +  2  /,  OQ  by  —  1  +  2 1. 


+  1      +2      +3      +4 


-2i 
Fig.  18 


A  symbol  of  the  for7n  x  -f  yi,  lohich  may  he  regarded  as 
representing  a  directed  line-segment  of  the  plane,  is  called  a 
complex  number.  This  complex  number  is  said  to  have  the 
positive  or  negative  numbers  x  and  y  as  its  components. 

Thus  a  complex  number  is  really  a  symbol  involving  a 
pair  of  ordinary  numbers  x  and  y.  The  numbers  which  we 
have  considered  so  far  might,  by  way  of  contrast,  be  called 
simple  numbers. 

We  may  regard  the  symbol  x  +  yi  as  a  description  of  a  path  -which 
leads  from  0  to  P.  Thus,  in  Fig.  18,  the  symbol  3  +  2 «  tells  us  to  start 
at  0,  to  go  3  units  in  the  direction  of   the   positive   ar-axis,  and    then 


36  THE   NUMBER   SYSTEM   OF   ALGEBRA     [Arts.  25,  26 

2  units  in  the  direction  of  the  positive  7/-axis.  If  the  x-axis  points  east 
and  the  y-axis  north,  we  may  regard  the  symbol  i  as  an  abbreviation  for 
north  and  —  i  for  south. 

25.  Equality  of  two  complex  numbers.  We  have  intro- 
duced the  complex  numbers  x  +  ^i  as  symbols  for  directed 
line-segments  of  the  plane,  and  we  have  agreed  that  two 
such  line-segments  are  to  be  regarded  as  equal  if  they  are 
of  the  same  length,  if  they  are  on  parallel  lines,  and  if  they 
have  tlie  same  sense.  The  symbols  for  two  equal  directed 
line-segments  will,  therefore,  be  identical,  since  equal  directed 
line-segments  have  their  corresponding  components  equal. 
(See  Fig,  16.)  It  is  also  evident  that  two  line-segments 
which  are  not  equal,  in  the  sense  of  the  above  definition, 
will  not  have  their  corresponding  components  equal. 

These  remarks  lead  us  to  the  following  definition  of 
equality  of  two  complex  numbers: 

Two  complex  numbers,  x  +  yi  and  x'  +  y'i,  are  said  to  he 

equal,  if  and  only  if  their  corresponding  components  are  equal, 

that  is,  if  ayid  only  if       ,  _         ,  _ 

X  —  X,  y  —  y. 

In  particular,  if  x  +  yi  =  0,  then  x=  0,  y  =  0. 

We  see,  therefore,  that  a  single  equation  between  two  com- 
plex numbers  implies  two  equations  between  their  components. 

The  discussion  in  Art.  23  shows  us  that  a  complex  num- 
ber may  also  be  said  to  have  a  inodulus  and  an  amplitude. 
It  follows  immediately  that  two  equal  complex  numbers  have 
equal  moduli  while  the  amplitudes  of  two  equal  complex  num- 
bers 7ieed  not  be  equal;  it  suffices  that  the  difference  between 
them  be  an  integral  multiple  of  360°. 

26.  Vectors,  vector  addition,  and  addition  of  complex 
numbers.  It  remains  to  justify  the  use  of  the  sign  +  in  the 
symbol  x  -f  yi.  If  we  write  the  number  x  alone,  we  may 
think  of  a;  as  a  directed  line-segment  OM  on  the  a;-axis. 
(See  Fig.  10.)  We  may,  a  little  more  concretely,  think  of 
OMnH  a  displacement  which  has  the  effect  of  moving  a  point 
from    0  to  M,  and  which  moves  every  other  point  of  the 


Art.  26] 


VECTORS,   VECTOR   ADDITION 


37 


+y 

X 

X 

f 

A 

y 

V 

0 

X 

U 

Fig.  1!) 


plane  through  a  distance  equal  to  OM  along-  a  line  parallel 
to  the  a;-axis.  Similarly  we  may  think  of  yi  as  the  symbol 
for  a  displacement  which  moves  every 
point  of  the  plane  through  a  distance 
equal  to  ON  along  a  line  parallel  to  the 
y-axis.  Now  let  us  think  of  these  two 
displacements  as  being  made  in  succession. 
The  first  displacement,  symbolized  by  x^ 
would  move  a  point  from  0  to  M',  the  second  displacement 
would  move  tliis  point,  which  is  now  at  iHf,  from  M  to  P. 
Since  the  same  result  would  have  been  obtained  if  the  point 
had  been  moved  directly  from  0  to  P  along  the  straight  line 
path  OP,  we  may  say  that  the  resultant  of  these  two  dis- 
placements is  that  one  which  is  represented  in  magnitude 
and  direction  by  OP.  Conversely,  the  two  displacements 
OiUfand  OiVare  called  rectangular  components  of  OP.  It  is 
customary  to  speak  of  the  resultant  of  two  displacements  as 
their  geometric  sum.  We  may  therefore  actually  regard 
X  +  yi  as  the  sum  (in  this  geometric  sense)  of  x  and  yi. 

The    actual    relation    between    the    magnitudes    of    the 
resultant  and  of  the  components  is  given  by 

oP=6m^^-on^, 

as  is  evident  from  the  figure. 

These  considerations  suggest  the  following  generalization. 
Let  OP  and  OQ  (Fig.  20)  represent  two  displacements. 
By  virtue  of  the  first  displacement  every 
point  of  the  plane  would  move  along  a  line 
parallel  to  (9P,  through  a  distance  equal 
to  the  length  of  OP,  in  the  sense  from  0 
toward  P.  The  directed  line-segment  OQ 
represents  in  similar  fashion  the  character- 
istic properties  of  the  second  displacement. 
It  is  clear  that  the  resultant  of  the  two  displacements  made 
in  succession  is  again  a  single  displacement,  represented  by 
Oi2,  where  OB,  is  that  diagonal  of  the  parallelogram  deter- 
mined by  OP  and  OQ  which  passes  through  0. 


38  THE   NUMBER   SYSTEM   OF   ALGEBRA        [Art.  26 

If  again  we  use  the  word  sum,  instead  of  resultant,  we 
may  say  that  the  sum  of  the  displacements  OP  and  OQ  is, 
the  displacement  OR. 

Let  us  now  represent  these  displacements  by  complex 
numbers,  z  and  z' .  If  the  components  of  OP  are  x  and  y^ 
and  those  of  0'^  are  x'  and  y',  we  have 

(1)  z  =  X  +  yi,  z'  =  x'  +  y'i. 

We  see  from  Fig.  20  that  the  components  of  OR  are  x  -\-  x' 
and  y  -\-  y' .     Consequently  we  obtain  the  following  result : 

If  the  ivord  sum,  ivlien  applied  to  two  displacements.,  he 
regarded  as  synonymous  with  the  word  resultant,  and  if  the 
complex  numbers  x  +  yi  and  x'  +  y'i  are  used  as  symbols  for 
two  displacements  ivhose  components  are  (a:,  ?/)  and  (x' ,  y'^ 
respectively,  then  the  sum  of  these  displacements  is  a  displace- 
ment ivhose  symbol  ivill  be  the  complex  number 

X  +  x'  +(y  i-  y')i. 

We  are  thus  led  to  define  x  -\-  x'  -\-  (^y  +  y'~)i  as  the  sum  of 
X  +  yi  and  x'  +y'i,  and  we  ivrite 

(2)  X  -{-  yi  +  (x'  +  y'i')=  x  +  x' +  Cy  +  y')i. 

There  are  many  instances  of  quantities  which  combine 
into  a  resultant  in  accordance  with  the  parallelogram  law 
illustrated  in  Fig.  20.  If  a  steamer  is  moving 
north  with  a  velocity  of  800  feet  per  minute, 
and  a  passenger  is  crossing  the  deck  walking 
eastward  at  the  rate  of  300  feet  per  minute, 
his  actual  velocity  in  space  may  be  obtained 
as,  in  Fig.  21,  by  laying  off  a  line-segment  OB, 
800  units  long  toward  the  north,  a  line-segment 
OA,  300  units  long  toward  the  east,  and  completing  the  paral- 
lelogram which,  in  this  case,  is  a  rectangle.  The  number  of 
units  in  OC  (obtained  by  measurement  or  b}'"  calculation) 
will  give  us  the  number  of  feet  per  minute  which  represents 
the  passenger's  speed  in  space,  and  the  direction  of  00, 
measured  by  the  angle  AOC,  will  give  us  the  direction  of  the 


Art.  26]  VECTORS,   VECTOR  ADDITION  39 

resulting  velocity.  Thus,  velocities  are  compounded  according 
to  the  parallelogram  law.  It  is  a  familiar  fact  that  the  resultant 
of  two  forces  attacking  a  body  at  the  same  point  is  also  found 
by  applying  the  parallelogram  law,  and  there  are  many  other 
instances  of  such  quantities. 

Directed  quantities,  such  as  displacements,  velocities,  and 
forces,  which  combine  in  accordance  with  the  parallelogram 
law,  are  called  vectors.* 

By  a  proper  choice  of  the  units  every  vector  may  be 
represented  by  a  directed  line-segment,  or,  what  amounts  to 
the  same  thing,  by  a  displacement.  We  shall,  from  now  on, 
use  the  word  vector  instead  of  directed  line-segment.  We 
may  then  say,  referring  again  to  Fig.  20,  that 

(3)  vector  OF  +  vector  OQ  =  vector  OR. 

Of  course  this  does  not  mean  that  the  length  of  OP  -j-  the 
length  oi  OQ  is  equal  to  the  length  of  OM.  In  fact  the  figure 
shows  that  the  length  of  OR  is  always  either  less  than  this 
sum  or  at  most  equal  to  it,  the  case  of  equality  presenting 
itself  only  if  the  Vectors  OP,  OQ,  and  OR  have  the  same 
direction.  This  simple  remark  may  be  formulated  alge- 
braically as  follows :  Let 

z  =  X  -\-  yi  and  z'  =  x'  +  y'i 

be  the  complex  numbers  which  correspond  as  symbols  to  the 
vectors  OP  and  OQ,  and  let  r  and  r'  be  their  respective 
moduli  or  absolute  values.  Then  r  and  r'  are  the  lengths  of 
these  vectors,  and 

r  =  Va-^  +  y^,  r'  =  Va:'^  +  y'^. 

It  is  customary  to  uae  the  symbol  \  x  -|-  yi  \  for  the  absolute 
value  or  modulus  of  a  complex  mimber.  The  observation  just 
made,  that  the  length  of  OR  is  at  most  equal  to  the  sum  of  the 

*  From  the  Latin  vector,  meaning  one  who  carries  or  conveys.  The  student 
should  observe  that  the  parallelogram  law  may  be  regarded  as  a  generalization 
of  the  geometric  definition  for  addition  wliich  was  given  in  Art.  18,  for  the  case 
where  the  line-segments  OA  and  O'A'  mentioned  in  Art.  18  are  not  situated  on 
the  same  or  on  parallel  lines. 


40 


THE   NUMBER   SYSTEM   OF   ALGEBRA        [Art.  27 


RJx-t-yi) 


lengths  of  OP  and  OQ,  may  therefore  be  written  as  follows; 

(4)  \x  +  i/i  +  x'  +  i/'i\  <\x  +  ^i\  -\-  \x'  +  y'i \ 

where  the  symbol  <  is  to  be  read  either  "  is  not  greater  than,'''' 
or  "is  less  than  or  at  most  equal  to.^^ 

It  is  apparent  from  Fig.  20  that  tlie  length  of  OM,  that  is, 
the  absolute  value  of  a;  +  i/i  +  a;'  +  y'i  is  exactly  equal  to 

(5)  -y/{x  +  xy  +  iy  +  i/'y. 

27.   Subtraction  of   vectors  and  complex  numbers.     If   we 

have  vector  OP  +  vector  0^=  vector  OR,  we  say  that 
Vector  0<^  =  vector  0^  —  vector  OP,  thus  defining  sub- 
traction of  vectors.  We  regard  the 
diagonal  OM,  and  one  of  the  sides 
OP  of  the  parallelogram  as  given  ; 
the  problem  of  subtraction  consists 
in  finding  the  other  side  OQ  of  the 
parallelogram.  The  geometric  so- 
lution is  so  simple  as  to  make  ex- 
plicit directions-  unnecessary.  See 
Fig.  22.  The  corresponding  for- 
mula for  the  subtraction  of  complex 
numbers  is  just  as  simple.    We  find 

(1)  (x  +  i/i)  -  (x'  +  y'i)  =  x  —  x'  +  (y  —  y')i, 

where  x  and  y  are  the  components  of  OR  the  minuend,  and  x', 
y'  are  the  components  of  the  subtrahend  OP.  The  absolute 
value  of  the  difference,  that  is,  the  length  of  OQ  or  PR,  is 


Fig.  22 


(2)       \x  +  yi  -  (2.-'  +  y'i-)  '  =  -^^(^x-x')''-\-(^y-y') 


'  \2 


EXERCISE   V* 

Plot  the  vectors  represented  by  the  following  complex  numbers,  and 
find  their  moduli  and  arguments  approximately  by  measurement: 

1.  1  +  1.  3.    -3  -  2«.  5.    -7. 

2.  -2  +  3i.  4.    5  -  2  I.  6.    8  i. 


*  An  arithmetical  solution  may  be  obtained,  where  a  graphic  solution  is  called 
for  in  these  examples,  by  those  students  who  liave  studied  Trigonometry. 


Art.  28]     MULTIPLICATION   OF   COMPLEX  NUMBERS  41 

Plot  the  following  vectors  whose  moduli  and  arguments  are  given. 
Find  their  components  approximately  from  the  figure  by  measurement, 
and  write  the  corresponding  complex  numbers. 

7.  ,=1,^  =  45°.  10.   r  =  4,  ^  =  315°. 

8.  7-  =  3,  ^  =  135°.  11.   r  =  5,d=  180°. 

9.  r  =  2.6=  225°.  12.    r  =  7,d  =  90°. 

What  must  be  the  values  of  x  and  y  in  order  that  the  following  equa- 
tions may  be  true? 

13.  X  +  ij  +  i(x  -  y)  =  2  +  4  i. 

14.  2  x  +  7  1/  +  i(S  X  -  2y)  =  -  3  -  i. 

Perform  the  following  operations  algebraically  and  graphically. 

15.  1  +  £  +  (2  +  3  0-  19-    7  +  8  I  -  (5  +  6  i). 

16.  1  +  t-  (2  +  3  0-  20.    -  1  +t  +  (2  +  3  0  -  (1+5  /). 

17.  3  -  5  I  +  (3  +  5  t).  21.    5.6  +  7.8  i  +  (  -  3.2  +  4.7  i). 

18.  3-5  i  -  (3  -  5  0-  22.    5.6  +  7.8  i  -  (3.2  +  4.7  i)- 

23.  A  horizontal  force  of  10  pounds  and  a  vertical  force  of  24  pounds 
are  acting  simultaneously  on  a  point.  Represent  the  resulting  force  as 
a  complex  number. 

24.  A  schooner  is  sailing  due  west  at  the  rate  of  6  miles  per  hour. 
A  sailor  is  crossing  the  deck,  from  south  to  north,  at  the  rate  of  3  miles 
per  hour.  Represent  the  resulting  velocity  of  the  sailor  as  a  complex 
number. 

25.  Two  forces  are  represented  by  the  complex  numbers  3  +  5  i  and 
—  4  +  6  I  respectively.  Find  the  complex  number  which  represents 
their  resultant.  If  the  x-axis  is  horizontal,  the  y-axis  is  vertical,  and  if 
the  unit  of  force  used  is  a  pound,  find  approximately,  by  a  graphic  solu- 
tion, the  magnitude  and  direction  of  the  resultant. 

28.  Multiplication  of  a  complex  number  by  a  positive  or 
negative  number.  //'  m  is  an//  positive  or  negative  number  or 
zero,  ive  define  the  product  m{x  +  yi)  to  he  equal  to  mx  +-  myi. 
If  m  is  positive,  the  corresponding  product  vector  will  be  m 
times  as  long  as  the  vector  x  +  yi,  but  its  direction  will  be 
the  same.  If  m  is  negative,  the  direction  of  the  product 
vector  will  be  opposite  to  that  oi  x  +  yi. 

This  definition  is  a  necessary  consequence  of  Art.  2fl  in  the  case 
where  m  is  a  positive  integer,  if  multiplication  by  a  positive  integer  be 
interpreted   as  repeated   addition.     This  same  definition    must  also   be 


42  THE   NUMBER   SYSTEM   OF   ALGEBRA        [Art.  29 

adopted  in  all  other  cases  if  we  wish  to  define  multiplication  of  a  com- 
plex number  by  m  iu  such  a  way  as  to  have  it  resemble  multiplication 
of  two  positive  or  negative  numbers  in  so  ftir  as  to  preserve  the  validity 
of  the  commutative  and  distributive  laws  iu  this  case  also. 

29.  Multiplication  of  a  complex  number  by  /.  Tlie  num- 
bers on  the  positive  rr-axis  were  denoted  by  +1,  +2,  +3, 
etc.  ;  those  on  the  positive  y-axis  by  +  i,  +2  i,  +  3  i,  etc. 
This  notation  suggests  the  possibility  of  regarding  3  i  as  a 
product  obtained  by  multiplying  i  by  3,  a  point  of  view 
which  is  entirely  consistent  with  Art.  28.  But  we  may  also 
think  of  3  i  as  the  product  obtained  by  multiplying  +  3  by 
i ;  in  fact  we  must  do  so,  if  we  wish  multiplication  by  i  to 
be  commutative,  that  is,  if  we  wish  3  x  i  to  be  equal  to  i  x  3. 
If  we  adopt  this  point  of  view,  we  see  that  multiplication 
by  i  of  any  vector  on  the  positive  a;-axis,  such  as  OM  in 
Fig.  23,  will  produce  a  vector  ON  of  equal  length  on  the 
positive  ?/-axis.  We  may  express  this  by  saying  tliat  mul- 
tiplication by  i  has  the  effect  of  turning  the  vector  OM 
through  a  right  angle  in  the  counter-clockwise  direction. 

If    we  wish    to    make   a  consistent   use  of   this   idea,  we 
should  agree  that,  to  multiply  ON  by  ^,  means  to  turn  ON 
+y  also  through  a  right  angle  in   the  counter- 

clockwise    direction.      But     this     operation 
would  convert  ON  into  OM',  a  vector  of  the 
M'    o      ~M^      same  length  as   OM  but  having  the  opposite 
Fig.  23  direction.     If  OM  is  of   length  x  units,  we 

shall  have  these  three  vectors  represented  by  the  following 
complex  numbers  (see  Fig.  23): 

OM  by  X,  ON  by  xi,  OM'  l^y  -  x. 

But,  since  OM'  was  also  obtained  from  ON  by  multiplica- 
tion with  2,  we  may  also  represent  OM'  by  the  symbol  (xi)i 
or  xi^,  and  the  two  symbols  for  OM' ,  x^  and  —a;,  will  be 
identical  if,  and  only  if  we  agree  that, 

(1)  ^^  =  ^•2  =  -l. 

Let  us  adopt  this  agreement,  and  let  us  further  agree  that, 
in  all  cases,  multiplication  of  a;  +  yi  by  i  shall  obey  the  cora- 


N' 


Art.  29]     MULTIPLICATION   OF  COMPLEX  NUMBERS  43 

mutative,  associative,  and  distributive  laws  of  multiplication, 
so  that 

(2)     i{x  +  yi)  =  ix  +  ii/i  =  ix  -{-yi^  =  ix  —  y  =  —  y  +  xi. 

Let  OP  (Fig.  24)  be  tlie  vector  whose  components  are 
a:  =  6>if,         y  =  MP. 

Let  OP'  be  a  vector  of  the  same  length  as  OP,  obtained  by 
turning  OP  through  an  angle  of  90°  in  the  counter-clockwise 
direction.  Then  OP'  will  make  the  same 
angle  witli  the  y-axis  which  OP  makes 
with  the  2:-axis,  and  the  triangles'  OP'M' 
and  0PM  will  be  equal.  Therefore  OM' 
=  x  and  P'M'  =  y.  The  components  of 
the  vector  OP'  are  M' P'  and  OM'  respec- 
tively.    If  we  denote  them  by  x'  and  ?/',  we  shall  have 

M'P'  =  x'  =  -y,       OM'=y'  =  +  x. 
so  that  the  complex  number  which  represents  OP'  will  be 

^  -\-  y'i  =  —  y  +  xi, 

which,  according  to  (2),  is  the  same  as  i(x-{-yi).     We  have 
therefore  the  following  result  : 

If  we  assume  that  the  operation  of  multiplying  a  complex 
number  x  +  yi  by  i  obeys  the  commutative,  associative,  and  dis- 
tributive laivs,  then  i^  ivill  be  equal  to  —\  and  such  a  multi- 
plication is  geometrically  equivalent  to  rotating  the  vector 
represented  by  x  -f-  yi  through  a  right  angle  in  the  counter-clock- 
wise direction. 

EXERCISE  VI 

Perform  arithmetically  and  graphically  the  following  multiplications. 
Multiply  : 

1.  2  i  by  3.  6.  (4  +  3  0'- 

2.  2i  by  -3.  7.  (-4  -  3  0'- 

3.  3-0  i  by  2.  8.  [2  +  5  /  -  (3  +  2  /)]  t'. 

4.  3  -  5  i  by  -  2.  9.  [(-  2  +  3  Oi]3. 

5.  3  -  5i  by  t.  10.  {[4  -I-  i  -(-2-30]t}(-2). 


44  THE   NUMBER   SYSTEM   OF   ALGEBRA     [Arts.  30,  31 

30.  Polar  form  of  a  complex  number.*  Let  r  and  6  denote 
the  modulus  and  amplitude  of  the  complex  quantity  x  +  yi. 
We  shall  then  have 

x=.  r  cos  6,  y  =  r  sin  6  (see  Art.  23,  Fig.  15), 

so  that  we  may  write 

(1)  X  -\-  yi  =  r(cos  6  -{-  i  sin  ^). 

Every  complex  number  can  be  written  in  the  form  (1),  wliich 
is  called  the  polar  form,  or  the  trigonometric  form  of  the 
complex  number.  The  polar  form  is  especially  convenient 
when  two  or  more  complex  numbers  are  to  be  combined  by 
multiplication  or  division. 

31.  Multiplication  of   two  complex   numbers.     Let  x  +  yi 

and  x'  +  iy'  be  any  two  complex  numbers,  and  let  us  form 

their  product,  assuming  that  the  commutative,  associative, 

and  distributive  laws  are  valid,  and  that  i"^  is  equal  to  —  1. 

These  are  the  same  assumptions  which  we  made  in  Art.  29. 

We  find  ,       , 

(x  +  yi^Qx  +  y'i^  =  xx'  +  xy  t  +  yix  +  yiy  i 

=  xx'  +  xy'i  +  x'yi  —  yy', 
or  finally 

(1)  (x  +  yi)  {x'  +  y'i)  =  xx'  -  yy'  +  (xy'  +  x'y)i. 

But  this  result  may  be  expressed  in  a  very  simple  form 
if  we  make  use  of  the  polar  form  of  the  complex  numbers 
involved. t  Let  r  and  r'  be  the  moduli,  and  6  and  6'  the 
amplitudes  of  the  two  complex  numbers  x  +  yi  and  x'  +  y'i 
respectively,  so  that  in  accordance  with  (1)  of  Art.  30, 

X  +  yi  =  r(cos  6  +  i  sin  ^), 
x'  -f  y'i  =  /(cos  6'  +  i  sin  6'). 
We  find 

(2)  (x  +  yi)  (x'  +  y'i)  =  rr'  [cos  0  cos  0'  -  sin  0  sin  0' 

+  ^(sin  0  cos  0'  +  cos  0  sin  0' )] . 

*  To  be  omitted  by  students  who  have  not  studied  Trigonometry 
t  Students  who  have  not  studied  Trigonometry  will  tind  an  alternative  treat- 
ment of  this  subject  in  fine  print  toward  tlie  end  of  tliis  article. 


Art.  31]     MULTIPLICATION   OF   COMPLEX   NUMBERS  45 

But,  according  to  the  addition  formulae,  of  the  trigonometric 
functions,  we  have 

cos  0  cos  6'  -  sin  6  sin  0'  =  cos(^  +  0')* 
sin  0  cos  0'  +  cos  0  sin  0'  =  sin((9  +  0'). 

Consequently  we  may  write  (2)  as  follows  : 

(3)      (x+7jiy(x'  +  i/'i)=rr'[co^{0  +  0')+  /sin  (0+0'-)], 

so  that  tlie  product  is  a  complex  quantity  whose  modulus  is 
rr'  and  whose  amplitude  is  ^  +  0'.  We  have  proved  the 
following  theorem  : 

Theorem  I.  If  the  associative,  commutative,  and  distrib- 
utive laws  of  multiplication  are  assumed  to  hold  for  all  com- 
plex numbers,  the  modulus  of  the  product  of  two  complex 
numbers  is  equal  to  the  product  of  the  moduli  of  the  factors,  and 
the  amplitude  of  the  product  is  equal  to  the  sum  of  the  ampli- 
tudes of  the  factors. 

We  may  express  this  in  a  different  way.  Let  us  speak  of 
X  +  yi  as  the  multiplicand,  and  of  x'  +  y'i  as  the  multiplier. 
According  to  (3),  the  product  will  be  represented  by  a  vector 
of  length  rr'  making  an  angle  0  +  0'  with  the  x-axis.  Now 
the  multiplicand  was  represented  by  a  vector  of  length  r 
making  an  angle  0  with  the  a:-axis.  We  shall  therefore 
obtain  the  product  vector  from  the  multiplicand  vector  by 
rotating  the  latter  through  an  angle  0'  and  stretching  it  in 
the  ratio  of  r'  :  1,  where  r'  and  0'  are  the  modulus  and  ampli- 
tude of  the  multiplier.     Thus  we  find 

Theorem  II.  The  product  vector  is  obtained  by  rotating 
the  multiplicand  vector  through  an  angle  equal  to  the  amplitude 
of  the  multiplier,  and  at  the  same  time  stretching  the  multipli- 
cand vector  in  the  ratio  r'  :  1,  ivhere  r'  is  the  modulus  of  the 
multiplier. 

*  Wilczynski-Slaught.  Plane  Trif/onometry  and  Applications  (Allyn  and 
Bacon),  p.  184.  Hereafter  this  book  will  be  referred  to  as  Plane  Trigonometry 
and  Applications, 


46  THE   NUMBER   SYSTEM   OF   ALGEBRA        [Art.  31 

In  Fig.  25,  OP  represents  the  multiplier,  OP'  the  multi- 
plicand, and   OQ  the  product.     Let   U  be  at  unit  distance 
from  0  on  the  positive  a:-axis.     Then  the 
angles  UOP  and  P'  OQ  are  both  equal  to 

6.  and 

OQ:  OP'  =  OP:  OU, 

since  0U=1,  OP  =  r,  OP'  =  /,  OQ  =  rr'. 
Consequently  the  triangles  OUP  and  OP'Q 
are  similar.  This  remark  may  be  formu- 
lated as  follows : 

Theokem  III.  If  each  of  Uvo  complex  numbers  he  repre- 
sented hy  a  vector^  the  vector  ivhich  represents  their  product  may 
he  found  hy  the  following   construction.      Construct  the  vector 

0  Z7,  of  unit  lengthy  on  the  positive  x-axis,  the  vector  OP  repre- 
senting the  multiplier  x  -\-  yi,  and  the  vector  OP'  representing 
the  multiplicand  x'  +  y'i.  Construct  a  triangle  OP'  Q  similar 
to  the  triangle  OUP.  Then  OQ  will  he  the  vector  which 
represents  the  product  (x  +  yi^(x'  +  y'i^- 

This  last  theorem  may  also  be  proved  without  making  use  of  Trigo- 
nometry. Equation  (1)  shows  that  the  product  of  x  +  iji  and  x'  +  y'i  is 
a  vector  OQ  whose  components  are 

x"  =  xx'  —  yy',     y"  —  xy'  +  x'y. 

Let  us  compute  the  length  |  OQ  \  of  this  product  vector  OQ.     We  have 

1  OQ  I  '^  =  x"'^  +  7/"2  =  {xx'  —  yy'y  +  {xy'  +  x'y)'^  =  x'^x''^  —  2  xyx'y'  +  y'^y''^ 

+  x'^y'^  +  2  xyx'y'  +  x'^y"^  =  x'^x''^  +  y^y'^  +  x'^y''^+  x''^y'^  =  x'^{x'^  +  y''^) 
+  y\x'^  +  y/'2)  =  (x2  +  2/2)  (x'2  +  2/'2), 
whence 


(4)  \0Q\=^  Vx2  +  y^  Vx'-^  +  y'-\ 

But  we  have 

(5)  \0P\=  Vx2  +  if,      I  OP'  I  =  Vx'2  +  y'\ 

SO  that  (4)  becomes 

(6)  1  OQ  I  =  I  OP  I  •  I  OP'  I, 

telling  us  that  the  modulus,  \  OQ  |,  of  the  product  is  equal  to  the  product  of 
the  moduli,  \  OP  |  and  \  OP'  |,  of  the  factors. 


Akt.  32]  DIVISION  OF  COMPLEX  NUMBERS  47 

To  prove  Theorem  III  we  now  refer  to  Fig.  25.     The  symbol  of  the 
vector  OU  h  1  +  0  •  i  and  we  have  |  0U\  =  1 ;  we  may  therefore  write 

(Q)  as  follows : 

^   ^  |(>(^1:10P|  =  \0P'\  :  I  Of/I. 

To  prove  that  the  triangles  OUP  and  OP'Q  are  similar,  it  will  suffice  to 
show  further  that 

(7)  \P'Q\:\UP\  =  \OP'\:\OU\. 

Making  use  of  the  notion  of  adding  of  vectors  (see  Art.  2G,  equation 
(3)),  we  have 

vector  OP  =  vector  OU  +  vector  UP, 
vector  OQ  —  vector  OP'  +  vector  P'Q, 

and  therefore, 

vector  UP  =  vector  OP  —  vector  OU  =  x  +  i/i  —  1, 

vector  P'Q  =  vector  OQ  —  vector  OP'  —  x"  +  y"i  —  {x'  +  y'i) 

=  x"  -  x'  +{u"  -  u')i, 
so  that 

(8)  \UP\=  \^{x  -  1)2  +  ,/,  \P'Q\=  V(a:"  -  x')2+(?/"  -  >j'y. 

The  components  x"  and  rj"  of  the  product  vector  were  given  by 

x"  =  xx'  —  yij',  y"  =  xy'  +  x'//, 
so  that      x"  —  x'  =  (x  —  l)x'  —  yy',  y"  —  y'  =(x  —  l)y'  +  x'y. 

Consequently  we  find  from  (8) 
I  P'Q  I 
=  V(x  -  1)V2  -  2  x'y'{x  -  \)y  +  ijhf^  +  (x  -  l)Y2+2  x'y'{x-\)y  +  x'-y''- 


=  \/(x  -  l)2(x'2  +  ?/'2)  +  y2(x'2  +  y'-'^)  =  Vx"-2  +  ^'2V(x  -  1)^  +  y% 

which  reduces  to 

(9)  I  P'Q  I  =  I  OP'  \  ■  \  UP\ 

on  account  of  (5)  and  (8).  But  (!>)  is  equivalent  to  (7)  since  |  0U\  =  1. 
Consequently  we  have  now  proved  Theorem  III.  Theorems  I  and  II  are 
merely  diffei-ent  ways  of  expressing  the  same  relations  and  are  immedi- 
ate consequences  of  Theorem  III. 

32.  Division  of  complex  numbers.  If  we  define  division  as 
the  operation  inverse  to  multiplication,  Theorem  I  of  Art.  31 
gives  us  the  following  result  without  any  calculation. 

Theorem.  The  modulus  of  a  quotient  h  equal  to  the  modu- 
lus of  the  dividend  divided  hy  the  modulus  of  the  divisor ;  the 


48  THE   NUMBER   SYSTEM   OF   ALGEBRA        [Art.  32 

amplitude  of  the  quotient  is  equal  to  the  amplitude  of  the  divi- 
dend minus  the  amplitude  of  the  divisor. 

We  may  liowever  prove  the  same  theorem  by  the  follow- 
ing calculation.     We  have 

r(cos  6  +  i  sin  6)   _    r(cos  6  -\-  i  sin  ^)(cos  6'  —  i  sin  6' ) 
r'(cos^'  +  zsin(9')  ~  ?7(cos  6'  +  i  sin  <9')(cos  6'  -  i  sin  9') 

_  r  cos  6  cos  6'  +  sin  6  sin  6'  +  i(sin  6  cos  6'  —  cos  6  sin  ^') 
r'  cos^  6  +  sin^  ^ 

But        cos  d  cos  6''  +  «in  6  sin  ^'  =  cos((9  -  (9'), 
sin  ^  cos  6'  -  cos  ^  sin  6'  =  sin(^  -  6'^, 
cos2  (9  +  sin2  e  =  \* 

Consequently  we  find 

,-1^      rCcos  ^  + « sin  ^)         r  r-        .n      /j^x   ,    •    •     ,a      m^-i 

<^1)    -77 ^,   .    .    -     ^,  =-[cos(^-6'^)  +  ^Sln(6'-^0], 

r  (cos  a'  +  z  sm  c'')      r 

and  this  formula  is  equivalent  to  the  above  theorem. 

If  the  complex  quantities  are  given  in  the  form  x  +  yi  and 
x'  +  y'^,  instead  of  in  the  polar  form,  their  quotient  may  be 
found  as  follows.      We  have 

,.-,.     X  +  yi  __  X  -{-  yi       x'  —  y'i  _  xx'  +  yy'  +  (x' y  —  xy')i 
x'  +  y'i  ~  x'  +  y'i      x'  -  y'i  ~  x'^  +  y'^ 

This  formula  shows  that  the  quotient  is  again  a  complex 
number  x"  +  y"i,  whose  components  are 

<-qN  ,,  ^xx'  +yy'       f,  ^x'y-xy' 

^^  x'^  +  y'^'  ^         x'^+y'^ 

Of  course  x"  and  y"  may  be  positive  or  negative.  The}'  are 
always  definite  numbers  obtained  from  x,  y,  :c',  and  ?/',  by  (3), 
unless  x'  and  y'  are  both  equal  to  zero,  that  is,  unless  the 
divisor  x'  +  y'i  is  equal  to  zero.  Therefore,  division  by  zero 
is  excluded  from  the  algebra  of  complex  numbers  just  as  ynuch 
as  from  the  algebra  of  real  numbers.     Two  complex  numbers 

*  Plane  Trigonometry  and  Applications,  pp.  148  and  186. 


AuT.  ;5_>]     OPERATIONS   WITH   COMPLEX   NUMBERS  49 

always  have  a  uniquely  determined  complex  number  for  their 
quotient  unless  the  divisor  is  equal  to  zero. 

The  laws  of  multiplication  and  division  of  complex  num- 
bers are  used  in  many  parts  of  applied  mathematics.  It  is 
not  feasible,  however,  to  present  any  of  these  applications 
in  this  book  because  tlie  preliminary  notions  needed  from 
Physics  are  so  complicated  as  to-  necessitate  a  lengthy  dis- 
cussion. 

EXERCISE  VII 

Perform  the  following  multiplicatiou.s  and  divisions  both  algebraically 
and  graphically : 

1.  (:5  +  5  0C2+3/).  6.  (3  +  5 /)  -  (2  +  3  0- 

2.  (3  -  5  /)  (2  -  ;}  0-  7.  (3  -  5  0  -  (2  -  3  i). 

3.  (-4  +  20(1  +  0.  8.  (-4  +  20-(l  +  0- 

4.  (2  +  3  i)-.  9.  1  ^  (2  +  3  0^. 

5.  (1  +  0-"  10.  1  -^(1  +  ly. 

Reduce  the  following  expressions,  in  whicli  n,  h,  c.  d,  etc.  represent 
positive  or  negative  numbers,  to  the  form  of  a  complex  number  A  +  Bi: 

11.  (a  +  Ih)(c  -  di).  14.    (rt  +  hi)  ^  (c  -  di). 

12.  (a  -  hi){c  +  di).  15.    {a  -  bi)  ^  (c  +  di). 

13.  (a  +  hi) (a  -  bi).  16.    (a  +  bi)  ^  (a  -  bi). 

Write  the  following  complex  numbers  in  the  form  x  +  yi:* 

17.  ;)(cos  30°  +  i  sin  30°).  21.    3(cos  90°  +  i  sin  90°). 

18.  3(cos  150°  +  /  sin  150°).  22.    2(cos  180°  +  i  sin  180°). 

19.  rj(cos  225°  +  i  sin  225°).  23.   3(cos  270°  +  i  sin  270°). 

20.  4(cos  0°  +  i  sin  0°).  24.   4(cos  360°  +  i  sin  360°. 

Write  the  following  complex  numbers  in  the  polar  form  : 

25.  1  +  /.  29.         I  +  I  y/■^  i.  33.        2  -  3  i. 

26.  -  1  +  /.  30.    -  f  +  f  V.^ ;.  34.  -  5  i. 

27.  -  1  -  /.  31.    -  f  -  f  \/3  I.  35.        5-4  i. 

28.  \-i.  32.    +  1  -  ^  >/3  i.  36.    -3  +  4  i. 

*  Examples  17-40  presuppose  knowledge  of  Trigonometry. 


50  THE   NUMBER   SYSTEM   OF   ALGEBRA        [Art.  33 

Perform  the  following  multiplications  and  divisions  by  first  reducing 
the  complex  numbers  concerned  to  their  polar  form. 

37.  (3  +  V3  0  (2  +  2  /) .  39.    (  -  J  +  i  2  v'3)8. 

38.  (1  +  0(2  -  2  V3  /)•  40.    (1  +  0  ^  (2  +  |V3  O- 

33.  Real  and  imaginary  numbers.  The  simple  numbers, 
that  is,  the  positive  and  negative  numbers  and  zero,  are  often 
spoken  of  as  real  numbers.  Of  course,  a  complex  number, 
x-\-yi^  reduces  to  such  a  real  number  when  «/=  0,  so  that  the 
real  numbers  are  included  among  the  complex  numbers.  Tliose 
complex  numbers  x  +  yi  in  which  y  is  not  equal  to  zero  are 
frequently  called  imaginary  numbers. 

The  reason  for  this  peculiar  nomenclature  is  not  hard  to 
understand.     No  real  number,  positive,  zero,  or  negative,  has 
a  negative  square.     Therefore,  in  the  domain  of  real  num- 
bers, an  equation  such  as 
(1)  a:2  =  -  1 

has  no  solution.  Mathematicians,  several  centuries  ago, 
found  it  convenient,  nevertheless,  to  treat  an  equation  like 
a;2  =  —  1  according  to  the  same  rules  as  were  used  for  equa- 
tions, such  as  x^  =  2,  which  do  have  real  solutions.  This  led 
to  such  symbols  as  V—  1.  It  was  clear  to  them,  of  course, 
that  the  symbol  V—  1  could  not  represent  a  real  number 
(that  is,  a  positive  or  negative  number),  and  it  seemed  to 
them  that  this  symbol  could  not  be  regarded  as  a  number  at 
all.  To  express  all  of  these  doubts  these  symbols,  whose 
significance  was  not  understood,  were  called  imaginary  num- 
bers, and  the  word  still  persists.  From  our  present  point  of 
view,  however,  any  complex  number  represents  a  real  thing, 
namely  a  vector  having  a  definite  length  and  direction.  And 
from  this  point  of  view  the  equation  x^  =  —  1  has  two  solu- 
tions. In  fact  we  found  in  Art.  29  that  the  vector  one  unit 
long  in  the  direction  of  the  positive  ^/-axis,  which  we  called 
i,  is  such  that  i^  =  —  1  and  it  is  apparent  that  (—  z')^  is  also 
equal  to  —  1.  Thus  we  see  that  the  equation  (1)  is  satisfied 
by 

''  X  =  I  QV  X  =  —  I. 


Arts.  34,  35]     VALIDITY   OF   FUNDAMENTAL   LAWS  51 

We  now  see  further  that  we  may  identify  our  symbol  i  with 
the  symbol  V—  1. 

The  symbol  V—  1  was,  from  the  old  point  of  view,  not 
a  real  number,  but  something  imaginary,  and  the  letter  i 
which  we  still  use  for  it  is  accounted  for  by  this  fact.  In 
the  technical  sense  we  shall  still  say  that  i  is  an  imaginary 
number,  meaning  that  it  is  not  a  positive  or  negative  number 
nor  zero.  But  it  is,  in  our  interpretation,  just  as  real  a  thing 
as  the  numbers  4-  1»  +  24,  or  —  3. 

Every  complex  number  is  of  the  form  x  +  yi-  We  shall, 
henceforth,  speak  of  a:  as  its  real  part  and  of  yi  as  its  imagi- 
nary part.  If  the  real  part  of  a  complex  number  is  zero,  the 
number  is  said  to  be  purely  imaginary. 

34.  Conjugate   complex   numbers.      Ttvo   complex  numbers 

such  as  ,      •       J  • 

X  +  yi  and  x  —  yi 

ivhose  real  parts  are  the  same,  and  for  which  the  coefficients  of  i 
are  numerically  equal  but  opposite  in  sign,  are  said  to  be 
conjugate. 

The  truth  of  the  following  statements  is  evident : 

The  sum  and  product  of  two  conjugate  complex  quantities  are 
both  real.     In  fact  we  have 

(1)  (x+ yi^  +  (x  —  yi)  =  'lx,     {x  +  yi)(x  -  yi)  =  x^-^  y^. 

The  difference  between  two  conjugate  complex  numbers  is 
purely  imaginary. 

(2)  {x  +  yi)  -i^-  yi)  =  2  yi. 

35.  Validity  of  the  fundamental  laws  for  complex  numbers. 
It  is  easy  to  verify  that,  with  the  exception  of  IV  and  VIII, 
the  fundamental  laws  of  Art.  2  are  all  satisfied,  if  the  symbols 
there  used  to  denote  positive  integei's  are  now  regarded  as 
standing  for  complex  numbers.  Laws  IV  and  VIII,  the 
monotonic  laws  of  addition  and  multiplication,  are  not  ap- 
plicable to  complex  numbers  for  the  simple  reason  that 
complex  numbers  cannot  be  arranged  in  a  simply  ordered 


62  THE   NUMBER   SYSTEM   OF   ALGEBRA        [Art.  36 

sequence  as  is  the  case  for  real  numbers.  In  other  words, 
the  fundamental  notions  "  greater  than  "  and  "  less  than  " 
have  no  simple  significance    for  complex  numbers. 

As  a  consequence  of  the  facts  just  noted,  we  may  manipu- 
late equations  involving  complex  numbers  according  to  the 
same  rules  as  though  the  numbers  concerned  were  real.  For 
the  monotonic  laws  have  nothing  to  do  with  equations  ;  they 
are  concerned  with  inequalities  only. 

36.  History  of  the  number  system  of  Algebra.  The  successive 
generalizations  which  have  enriched  the  number  concept,  beginning  with 
the  primitive  idea  of  a  positive  integer  and  leading  up  to  the  general  notion 
of  a  complex  number,  were  not  obtained  by  sudden  inspiration,  but  as  a 
result  of  the  painstaking  work  of  mathematicians  throughout  thousands 
of  years.  Even  our  present  method  of  writing  numbers,  the  conven- 
ience of  which  can  be  appreciated  only  by  contrasting  it  with  the  clumsy 
methods  used  by  the  Greeks  and  Romans,  was  the  result  of  a  long  proc- 
ess of  evolution.  The  most  essential  feature  of  this  system,  its  positional 
character,  was  rendered  possible  only  by  the  invention  of  a  symbol  for 
zero.  It  is  usually  conceded  that  the  Hindu  mathematicians  of  the  sixth 
century  a.d.  took  this  step,  although  there  seems  to  be  some  evidence 
that  the  Babylonians  also  had  a  symbol  for  zero.  The  characters  0,  1,2, 
etc.  which  we  use  nowadays  are  also  supposed  to  be  of  Hindu  origin. 
We  usually  speak  of  them  as  Arabic  figures,  because  they  were  trans- 
mitted to  the  nations  of  western  Europe  through  the  Arabs,  who  had 
previously  received  them  from  the  Hindus. 

The  oldest  mathematical  manuscript  with  which  we  are  acquainted, 
the  so-called  Rhind  papyrus,  written  by  the  Egyptian  Ahmes  about  1700 
B.C.,  contains  many  calculations  which  involve  fractions.  P^xcept  for  the 
fraction  2/3  all  of  the  fractions  used  by  Ahmes  were  unit  fractions,  that  is, 
such  fractions  as  1/2,  1/3,  1/4,  etc.,  whose  numerators  are  equal  to  unity. 
The  Babylonian  astronomers  introduced  the  system  of  sexagesimal  frac- 
tions, which  was  far  superior,  for  purposes  of  practical  reckoning,  to  the 
system  of  unit  fractions  used  by  Ahmes.  In  this  system  every  unit 
is  divided  into  sixty  equal  parts,  each  of  these  is  divided  into  sixty 
smaller  parts,  etc.  The  sexagesimal  system  was  adopted  by  Ptolemy  of 
Alexandria  (about  1.50  a.d.)  in  his  famous  treatise  on  Astronomy,  the 
Abnar/esi,  and  remained  in  general  use  for  scientific  purposes  until  the 
sixteenth  century,  when  it  was  replaced  by  tlie  modern  decimal  system. 
We  still  have  important  traces  of  the  sexagesimal  system  in  our  reckoning 
of  time  and  angles.  Thus  we  divide  an  hour  into  sixty  minutes,  and  a 
minute  into  sixty  seconds;  we  also  divide  a  degree  of  arc  into  sixty 
minutes,  and  a  minute  of  arc  into  sixty  seconds. 


Art.  36]         HISTORY   OF   THE   NUMBER   SYSTEM  63 

As  was  mentioned  in  Art.  15,  the  discovery  of  the  distinction  between 
rational  and  irrational  quantities  is  usually  ascribed  to  Pythagoras. 
But  Pythagoras  and  the  Greek  mathematicians  who  followed  him  made 
this  distinction  in  an  exaggerated  fashion.  They  refused  to  regard  irra- 
tional quantities  as  numbers  at  all,  and  did  not  introduce  them  into  their 
Arithmetic  and  Algebra.  Further  progress  in  this  direction  was  made 
possible  only  by  the  Hindus,  especially  by  Bhaskara  (1114  a.d.).  The 
Hindus  did  not  perhaps  appreciate  the  fundamental  character  of  this  dis- 
tinction as  clearly  as  the  (ireeks,  but  they  showed,  by  actual  trial,  that  it 
was  possible  to  use  irrational  numbers,  and  to  calculate  with  them  ac- 
cording to  the  same  rules  that  hold  for  rational  numbers.  It  was  re- 
served for  the  nineteenth  century  to  justify  the  procedure  of  the  Hindus 
by  a  strictly  logical  theory  of  irrational  numbers  such  as  would  have  been 
acceptable  to  the  Greeks.  This  theory  is  due  to  Uedekxnd,  Cantor, 
and  Weierstrass. 

Negative  numbers  also  seem  to  have  appeared  first  among  the  Hindus. 
Much  later  they  gradually  forced  themselves  also  upon  the  attention  of 
occidental  mathematicians.  Thus,  we  find  that  Leonardo  of  Pisa  or 
Fibonacci  (1180-12.50)  accepts  negative  solutions  of  an  equation  in  all 
cases  where  these  negative  numbers  are  capable  of  being  interpreted  as 
debts.  The  great  Italian  mathematicians  of  the  Renaissance,  Cardano, 
BoMHELLi  (sixteenth  century  a.d.),  and  others  followed  Leonardo  in 
this  practice.  They  also  began  to  use  imaginaries  in  their  calculations, 
although  they  regarded  them  with  much  suspicion.  The  geometric  repre- 
sentation of  positive  and  negative  numbers  as  directed  line-segments  on  a 
directed  line  was  contained  siibstantially  in  the  famous  Ge'ometrie  of 
Descartes  (1596-1650),  one  of  the  founders  of  what  we  now  call  Ana- 
lytic Geometry.  As  a  result  of  this  interpretation,  it  was  generally 
recognized  that  negative  numbers  had  established  their  citizenship  in  the 
republic  of  numbers. 

In  the  same  way,  complex  numbers  were  regarded  as  mere  phantoms 
and  of  no  real  significance,  until  an  adequate  geometric  representation 
was  found  for  them.  This  was  accomplished  independently  in  1797  by 
Caspar  Wessel,  a  Dane  ;  in  1806  by  J.  R.  Argand,  a  French  mathema- 
tician ;  and  finally  in  1831  by  C.  F.  Gauss.*  The  influence  of  the  latter, 
who  was  probably  the  greatest  mathematician  of  the  nineteenth  century, 
and  the  important  applications  which  he  made  of  complex  numbers,  finally 
established  the  complex  numbers  in  the  place  which  they  have  occupied 
ever  since,  as  the  only  completely  adequate  number  system  of  Algebra. 

*  This  is  the  representation  upon  which  we  have  based  our  theory  of  complex 
numbers  in  Arts.  23-.34,  and  is  frequently  referred  to  as  the  Argand  diagram. 
This  representation  defines  a  one-to-one  correspondence  between  the  points  of  the 
plane  and  the  values  of  the  complex  variable  x  +  yi.  Consequently,  a  plane,  to 
each  of  whose  points  there  corresponds  in  this  way  a  complex  number,  is  often 
called  the  plane  of  the  complex  variable,  or  the  Gauss  plane. 


CHAPTER   II 

LINEAR   FUNCTIONS   AND    PROGRESSIONS 

37.  Constants  and  variables.  When  we  have  solved  a 
numerical  problem  by  the  methods  of  elementary  Arithmetic, 
we  usually  recognize  that  there  are  other  numerical  problems 
of  the  same  kind  which  may  be  solved  by  applying  the  same 
methods.  This  leads  to  the  formulation  of  certain  rules, 
which  state  that  in  certain  problems  certain  numbers  must 
be  added,  subtracted,  multiplied,  or  divided.  Algebra,  by 
introducing  the  notion  that  any  number  whatever  may  be 
represented  by  a  letter,  such  as  a,  6,  <?,  x,  or  y,  enables  us  to 
formulate  such  rules  in  a  very  concise  manner.  Thus,  in 
Algebra,  we  focus  our  attention  not  on  a  single  numerical 
problem,  but  upon  a  whole  class  of  similar  problems.  The 
possibility  of  solving  all  examples  of  a  certain  class  at  once, 
except  for  the  mere  numerical  operations  involved,  consti- 
tutes one  of  the  advantages  of  Algebra  over  elementary 
Arithmetic. 

Thus,  in  Algebra,  the  rules  of  Arithmetic  are  replaced 
by  algehraic  formulce.  In  such  formulse  there  are  usually 
present  some  symbols  (letters)  which  are  intended  to  repre- 
sent fixed  numbers,  and  other  symbols  to  which  various 
values  may  be  assigned.  Thus  in  the  formula  for  the  area 
of  a  circle 
(1)  A  =  7rr2, 

the  number  tt  (approximately  equal  to  3.14159)  is  always 
the  same.  But  formula  (1)  enables  us  to  calculate  the  area 
of  any  circle.  If  r  represents  the  number  of  length  units 
(inches  for  instance)  in  the  radius  of  the  circle,  then  7rr^  will 
represent  the  number  of  area  units  (square  inches)  in  the 

64 


Art.  37]  CONSTANTS   AND   VARIABLES  56 

area.  Thus  the  letter  r,  in  (1),  may  be  thought  of  as 
representing  any  number  whatever,  afd  we  therefore  speak 
of  it  as  a  variable.  We  may  even  think  of  r  as  being  vari- 
able in  a  more  concrete  fashion.  Let  us  consider  a  circular 
disk  made  of  metal  placed  in  a  room  of  variable  temperature. 
As  the  temperature  rises,  r  will  increase,  tt  will  remain 
constant,  and  A  will  increase  with  r  according  to  the  law 
given  by  equation  (1). 

A  symhol  which,  in  a  given  class  of  problems,  is  regarded  as 
representing  a  fixed  number,  entirely/  incapable  of  change,  is 
called  a  constant. 

A  symbol  ichich,  in  a  given  class  of  problems,  may  be  regarded 
as  assumi7ig  different  values  is  called  a  variable. 

Thus,  in  (1),  A  and  r  are  variables,  w  is  a  constant. 

Equation  (1)  contains  two  variables,  A  and  r,  each  of 
which  is  capable  of  assuming  different  values.  But  this 
equation  states  a  relation  between  the  variables,  such  that 
whenever  the  value  of  r  is  given,  the  value  of  A  is  deter- 
mined. We  express  this  fact  by  saying  that  ^  is  a  function 
of  r,  in  accordance  with  the  following  definition: 

If  two  quantities,  x  and  y,  are  so  related  that  to  definitely 
assigned  values  of  x  there  correspond  definite  values  of  y,  then  y 
is  said  to  be  a  function  of  x;  the  variable  x  is  called  the  inde- 
pendent variable  or  argument,  and  y  is  called  the  dependent  vari- 
able or  function. 

Ordinarily  when  ?/  is  a  function  of  x,  a  change  in  x  will  produce  a  cor- 
responding change  in  ?/.  But  it  does  not  contradict  the  above  definition 
to  include  the  case  when  //  is  a  constant  as  an  instance  of  a  functional 
relation  between  x  and  ij.  In  that  case  the  values  of  y,  which  correspond 
to  different  values  of  x,  are  all  equal. 

The  distinction  between  constants  and  variables  may  also  be  looked 
upon  as  one  of  degree  rather  than  of  kind.  A  variable  x  may  be  capable 
of  assuming  all  possible  real  and  complex  values,  or  only  all  possible  real 
values,  or  only  some  real  values,  or  in  extreme  cases,  only  one  real  value. 
It  is  therefore  often  cdnvenient  to  regard  a  constant  as  a  special  case  of  a 
variable,  namely  that  special  case  where,  on  account  of  the  nature  of  the 
problem,  the  variable  can  assume  only  one  value.     Moreover  we  shall 


56  LINEAR   FUNCTIONS   AND    PROGRESSIONS     [Art.  38 

often  have  occasion  to  regard  one  of  the  quantities,  x  or  y,  as  unknown 
and  we  may  not  be  able  to  state  beforehand  whether  this  unknown 
quantity  may  assume  more  than  one  value  or  indeed  any  value  at  all.  It 
will  therefore  be  wise  to  treat  this  unknown  quantity  as  variable,  until 
the  solution  of  our  problem  has  taught  us  whether  it  is  a  variable  in  the 
strict  sense  of  the  word  or  a  constant. 

38.   Variation.     A  very  simple  and  important  instance  of 
a  functional  relation  between  two  variables  is  furnished  by 
the  equation 
(1)  y  =  mx, 

where  m  is  a  constant,  and  x  and  y  are  variables. 

Relations  of  this  form  occur  very  frequently  in  ordinary  life.  Thus 
if  m.  repi'esents  the  price  of  one  pound  of  sugar  and  x  the  number  of 
pounds  bought,  y  =  mx  will  be  the  cost  of  x  pounds.  If  m  is  the  rent  of 
a  house  for  one  month,  y  =  mx  will  be  the  rent  due  at  the  end  of  x 
months. 

When  two  variables,  x  and  z/,  are  connected  by  a  relation  of 
the  form  y  =  mx,  where  m  is  a  constant,  y  is  said  to  vary  as  x. 

The  same  relation  may  also  be  expressed  by  saying  that  y 
is  proportional  to  x,  on  account  of  the  following  theorem  : 

If  y  varies  as  x,  any  two  values  of  y  are  to  each  other  as  the 
corresponding  two  values  of  x. 

Proof.  Let  x^  and  x^  denote  any  two  values  which  x 
may  assume,  and  let  y^  and  y^  represent  the  corresponding 
values  of  y.*     Then  we  must  have 

yi  =  mx^,  y^  =  mx^, 


whence 

y±= 

mx^ 

X- 

y% 

mx^ 

X. 

as  was  to  be 

pr 

oved. 

*  The  subscript  notation  Xi,  x.2  is  very  convenient  and  will  be  used  frequently 
in  this  book.  We  read  Xj  and  x^,  x-sub-one  and  x-suh-tioo  respectively,  x^  and 
x-i  represent  any  two  values  which  x  may  assume.  The  a;-part  of  the  notation 
reminds  us  of  the  fact  that  each  of  these  numbers  is  a  value  of  x.  The  subscript 
or  index  indicates  that  we  are  thinking  of  ^,  first  and  a  second  value  of  x.  Thus  r-< 
does  not  mean  the  same  thing  as  x'^  {x  square) . 


Art.  38]  VARIATION  57 

The  constant  m  which  occurs  in  (1 )  is  called  the  constant 
of  variation,  or  the  factor  of  proportionality. 

The  symbol  ac  is  sometimes  used  to  denote  variation. 
Thus  y  Qc  a;  is  read  y  varies  as  x.  The  use  of  this  symbol  is, 
however,  gradually  becoming  obsolete. 

If  a  variable  y  varies  as  1/a;,  that  is,  if 

m 

X 

y  is  said  to  vary  inversely  as  x.     By  way  of  contrast  we  say 
that  y  varies  directly  as  x,  if 

y  =  mx. 

A  variable  z  is  said  to  vary  as  x  and  y  jointly  if 

z  =  mxy ; 

it  is  said  to  vary  directly  as  x  and  inversely  as  y,  if 

X 

y 

If  we  know  in  the  first  place  that  y  varies  as  x  and,  in  the 
second  place,  what  value  of  y  corresponds  to  a  given  specific 
value  of  a;,  we  can  determine  the  constant  of  variation  and  write 
down  the  relation  between  x  and  y  as  an  equation  of  the 
form  (1)  with  a  known  value  of  m. 

Thus,  let  y  vary  as  x,  and  let  //  =  30  when  x  =  2.     Then  we  have 

y  —  mx,  30  =  ?n  •  2, 
whence  m  =  15,     y  =  15  x. 

EXERCISE  VIII 

In  the  following  examples  determine  the  constant  of  variation  and 
state  the  indicated  relation  as  an  eqnation  : 

1.  //  varies  as  .?;,  and  //  =  16  when  .r  =  2. 

2.  y  is  proportional  to  .r.  and  //  =  (54  when  .r  =  4. 

3.  -1  varies  as  r-  and  A  =  tt  when  r  =  1. 

4.  V  varies  inversely  as  p  and  r  =  1  when  p  =  \. 

5.  z  varies  as  x  and  y  jointly  and  z  =  o  when  x  =  2  and  y  =  3. 


58      LINEAR   FUNCTIONS   AND  PROGRESSIONS     [Arts.  39, 40 

6.  z  varies  directly  as  x  and  inversely  as  y,  and  ;:  =  10  when  x  =  3 
and  ^  =  -i. 

7.  Prove  the  theorem  converse  to  the  theorem  of  Art.  38.  If  two 
variables,  x  and  y,  are  so  related  that  yi  :  y-^  =x-^ :  x^,  where  (xj,  y^)  and 
(xj,  y^)  are  ajiy  two  sets  of  corresponding  values  of  x  and  y,  then  y 
varies  as  x. 

8.  Prove  the  theorem  :  If  z  depends  only  on  x  and  y,  and  if  z  varies  as 
X  when  y  remains  constant,  and  varies  as  y  when  x  remains  constant, 
then  z  varies  as  x  and  y  jointly  when  both  x  and  y  vary. 

39-  Application  to  concrete  problems.  In  most  applications 
the  numbers  x  and  y  represent  the  numerical  measures  of 
concrete  quantities,  such  as  lengths,  areas,  volumes,  etc.,  and 
therefore  depend  upon  the  unit  of  measu-rement. 

Thus,  if  X  represents  the  numerical  measure  of  a  length,  we  may  have 
x  =  4,  or  48,  or  4/3,  or  4/5280  for  the  same  distance,  according  as  the 
unit  employed  is  a  foot,  an  inch,  a  yard,  or  a  mile. 

Consequently,  if  x  and  y  are  the  numerical  measures  of  two 
concrete  quantities^  and  if  y  varies  as  x,  the  value  of  the  factor 
of  proportionality  m  tvill,  in  general,  depend  upon  the  units 
employed. 

The  factor  of  proportionality  will  be  independent  of  the  units,  only  if 
X  and  y  are  the  measures  of  quantities  of  the  same  kind.  In  such  cases 
m  will  be  an  abstract  number. 

Many  of  the  measurable  quantities  which  occur  in  Physics 
and  Chemistr}',  and  in  the  applications  of  these  sciences  to 
Engineering,  may  be  expressed  in  terms  of  length.,  time,  and 
mass.  We  shall  now  discuss,  very  briefly,  these  fundamental 
notions  and  some  others  derived  from  them,  so  that  the  con- 
crete examples  involving  variation,  which  follow  in  Exercise 
IX,  may  become  endowed  with  some  real  significance. 

40.  Measurement  of  length.  Let  us  use,  as  standard  of 
length,  a  rod  one  meter  in  length  divided  into  one  hundred 
equal  parts  called  centimeters.  If  there  are  no  smaller 
divisions  on  the  scale,  such  a  standard  will  enable  us  to 
measure   a   given   distance   to    the   nearest   centimeter.     A 


Art.  40]  MEASUREMENT  OF  LENGTH  69 

vernier*  may  be  used  to  measure  the  fractional  part  of  the 
centimeter  which  should  be  added. 

Fig.  26  represents  two  scales  5  and  V,  which  are  capable  of  sliding 
along  each  other.  Let  us  think  of  the  upper  scale  .S  (wliich  is  divided 
into  eeutimeters)  as  fixed,  and  the  lower  (the  vernier  scale  V)  as  movable. 
If  we  wish  to  measure  the  distance  between  two  points  M  and  M'  we 
may  first  bring  the  vernier  scale  into  contact  with  M  and  read  the 
division  of  the  upper  scale  which  is  opposite  to  the  zero  point  or  index  of 
the  vernier,  and  then  bring  the  vernier  scale  into  contact  with  M'  and 
read  the  division  opposite  to  the  index  of  the  vernier  when  it  occupies 
this  second  position.  The  difference  between  the  two  readings  will  be 
the  required  distance  in  centimeters.     In  Fig.  20  the  index  of  the  vernier 


BD 

I   3 


Fig.  26 

points  to  a  place  B  on  the  upper  scale  between  2  and  '•].  We  wish  to 
find  out  how  many  tenths  of  a  centimeter  there  are  in  the  distance  be- 
tween the  points  marked  2  and  B  on  the  upper  scale.  The  divisions  on 
the  vernier  scale  are  closer  together  than  those  on  the  principal  scale,  each 
of  the  intervals  being  only  9/10  of  a  centimeter  in  length.  Consequently 
the  division  marked  1  on  the  vernier  will  be  closer,  by  1/10  of  a  centimeter, 
to  the  division  3  on  the  upper  scale,  than  was  the  zero  point  of  the  vernier 
to  the  division  2  of  the  upper  scale.  The  divisions  of  the  vernier  scale 
approach  the  preceding  division  of  the  upper  scale  at  the  rate  of  1/10  of 
a  centimeter  per  division.  Now  (in  Fig.  26)  the  division  7  of  the  vernier 
coincides  with  a  division  of  the  upper  scale,  so  that  it  takes  seven  of 
these  approaches  to  reduce  this  distance  to  zero.  Therefore  the  distance 
from  2  to  B  on  the  upper  scale  must  be  ■^^J  of  a  centimeter  or  7  milli- 
meters. Thus  the  reading  is  2.7  centimeters  when  the  vernier  is  in 
contact  with  M. 

In  order  to  formulate  this  theory  in  general,  let  the  prin- 
cipal scale  be  w  + 1  units  long,  and  let  the  vernier  scale,  n 
units  long,  be  divided  into  n  +  1  equal  parts  so  that  each 

*  So  named  after  Pierre  Vernier  (1.580-1637),  born  at  Ornans  in  France. 
The  instrument  is  also  frequently  called  a  nonius  after  Pedro  Nunez  (1492-1577), 
a  Portut^uese,  professor  of  mathematics  at  Coimbra,  whose  invention  was  based  on 
the  same  principle,  but  was  not  quite  so  convenient. 


60  LINEAR  FUNCTIONS   AND   PROGRESSIONS     [Art.  40 

division  of  the  vernier  scale  has  a  length  of  nj (n  -f  1)  units. 
(In  our  example  w  =  9,  and  the  unit  is  one  centimeter.) 
I^et  the  zero  point  (index)  of  the  vernier  fall  between  the 
divisions  a  and  a  -|-  1  of  the  principal  scale.  (In  our  ex- 
ample a  =2.)  The  scale  reading  will  then  be  a  4- a;  where 
a:  is  a  proper  fraction  which  is  to  be  found.  (In  Fig.  26,  x 
is  the  distance  2  B^  and  1  —  a;  is  the  distance  3  ^.)  Suppose 
that  the  division  h  of  the  vernier  scale  is  the  first  one  which 
coincides  with  a  division  of  the  principal  scale,  at  the  point 
(7  of  Fig.  26.  (In  our  example  A:=  7.)  Then  the  distance 
^(7  contains  k  vernier  units,  so  that 

(1)  BC=    ^^ 


w4-  1 


On  the  other  hand,  there  will  be  A;  —  1  complete  divisions  of 
the  principal  scale  between  B  and  (7  (in  our  example 
A-  — 1  =  6),  and  a  fractional  part  of  a  division,  marked  BB  in 
Fig.  26,  which  is  equal  to  \  —  x.  Therefore  we  may  also 
write 

(2)  BC^X-x  +  h-X^k-x. 

If  we  equate  the  two  values,  (1)  and  (2),  of  BG^  we  find 

kn    _-, 

—  —  K  —  x, 

w  +  1 

whence     hn  =  (w  +  V)k—  (n  +  l^x  =  nk  +  k  —  (^n  -\-  l)a:, 
or 

(3)  x=     ^ 


n  +  1 ' 

a  result  which  verifies  the  conclusion  drawn  in  the  illustrative 
example.  We  obtain  the  following  general  theorem  about 
verniers  : 

Let  the  principal  scale  be  n  +  1  units  long^  and  let  the  vernier 
scale  of  length  n  units  be  divided  into  n  +  1  equal  parts.  If 
the  index  of  the  vernier  falls  betwee/i  tivo  divisions.,  a  and  a  +  1, 
of  the  principal  scale,  and  if  the  division  k  of  the  vernier  is  the 


Aim.  n]  ij-:x(;rii,  area,  and  volume  61 

first  which  coincides  ivith  a  dividon  of  the  principal  scale,  then 
the  index  of  the  vernier  points  to  the  place 

h 

a  + 


M  +  1 

on  the  principal  scale. 

Thus  a  veniitT  may  1>p  constructed  to  enable  us  to  read  any  kind  of 
subdivisions  not  indicated  on  tlie  principal  scale.  Moreover,  the  two 
scales  may  be  circular;  the  ]n-inciple  remains  uncihanged.  Consequently 
verniers  may  also  be  employed  in  tlie  measurement  of  angles,  and,  in  fact, 
this  is  one  of  their  most  important  applications. 

41.  Length,  area,  and  volume.  The  result  of  measuring  a 
length  is  expressed  by  saying  that  the  required  distance  d 
contains  x  units  of  length,  or  briefly 

(1)  d  =  x-  L, 

if  L  denotes  the  unit  of  length.  In  this  equation  x  is  an 
abstract  number  and  the  product  xL  represents  a  length  only 
on  account  of  the  presence  of  the  factor  L. 

Areas  and  volumes  are  usually  determined,  not  by  direct 
measurement,  but  by  calculation  from  certain  lengths.  As 
unit  of  area  we  usually  select  a  square,  represented  by  I?, 
whose  side  is  the  unit  of  length ;  and  the  numerical  measure 
of  any  area  is  an  abstract  number  which  tells  us  how  many 
times  the  unit  area  is  contained  in  the  given  one.  Thus, 
any  area  may  be  regarded  as  a  product  of  the  form 

(2)  A=x-U 

where  x  is  an  abstract  number  and  where  X^,  the  unit  of 
area,  is  a  square  whose  side  is  X,  the  unit  of  length. 

Similarly,  any  volume  Fmay  be  regarded  as  a  product, 

(3)  V=x-L\ 

of  an  abstract  number  x  and  a  unit  of  volume  i^,  a  cube 
whose  side  is  equal  to  the  unit  of  length. 

Formulae  (1),  (2),  (3)  may  be  summed  up  by  stating 
that  the  dimensions  of  a  length,  area,  and  volume  are  i,  L\ 
L^  respectively. 


62     LINEAR  FUNCTIONS   AND   PROGRESSIONS     [Arts.  42-44 

These  dimensional  symbols  make  it  very  easy  to  change 
from  one  unit  of  measurement  to  another. 

Thus  we  may  convert  an  area  of  3  square  feet  into  square  inches  as 
follows  :         g  ^^^  ,^2  ^  g^^2  in.)2  =  3  x  144  (in.)^  =  432  (in.)^. 

42.  Time.  The  fundamental  and  natural  unit  of  time  is 
the  day.  It  is  determined  by  the  uniform  rotation  of  the 
earth  around  its  axis.  The  shorter  units,  hours,  minutes, 
and  seconds,  are  measured  by  the  use  of  clocks,  whose  ap- 
proximately uniform  motion  is  guaranteed  by  their  mechani- 
cal construction.  We  shall  use  the  symbol  T  to  represent  a 
unit  of  time. 

43.  Mass.  If  two  bodies  exactly  balance  each  other  when 
placed  on  the  two  scales  of  a  balance  with  equal  arms,  they 
are  said  to  have  equal  masses.  If  any  body  be  chosen  as  a 
standard,  and  if  it  takes  m  of  these  standard  bodies  to  bal- 
ance another  body,  the  latter  is  said  to  contain  m  mass-units, 
or  else  its  mass  is  said  to  be  equal  to  m.  The  unit  of  mass 
most  frequently  used  in  scientific  measurements  is  called  a 
gram.  A  gram  is  the  mass  of  a  cubic  centimeter  of  water 
at  a  temperature  of  4°  Centigrade.  1000  grams  or  1  kilo- 
gram is  equivalent  to  2.2046  English  pounds.  We  shall  use 
the  symbol  Mior  tlie  unit  of  mass. 

44.  Density  and  Specific  Gravity.  The  mass  of  a  body  de- 
pends upon  the  nature  of  its  material  and  its  volume.  A 
body  is  said  to  be  homoc/eneous  if  all  of  its  parts  are  exactly 
alike.  If  a  unit  volume  of  such  a  homogeneous  body  con- 
tains p  mass  units,  v  unit  volumes  will  contain  pv  mass 
units,  so  that  we  shall  have 

(1)  m  =  pv, 

if  m  denotes  the  total  mass  and  v  the  complete  volume  of  the 
body. 

The  number  p,  which  expresses  the  number  of  mass  units  in  a 
unit  volume,  and  ivhich  is  different  for  different  bodies,  is 
called  the  density  of  the  body. 


Art.  45]  VELOCITY  63 

Equation  (1)  expresses  the  fact  that  the  mass  of  a  homo- 
geneous body  varies  as  its  volume  ;  the  factor  of  proportion- 
ality in  this  case  is  the  density.  Since  we  may  write  (1)  as 
follows 

(2)  P  =  -, 

and  since  the  symbols  for  unit  of  mass  and  unit  of  volume 
are  iHf  and  L^  respectively,  we  find  the  symbol  M/L^  for  the 
unit  of  density. 

Let  us  use  the  centimeter  as  unit  of  length  and  the  gram  as  unit  of 
mass.  A  cubic  centimeter  of  water  has  (by  definition  of  the  gram)  the 
mass  of  one  gram.  Therefore  the  application  of  equation  (2)  to  a  cubic 
centimeter  of  water  teaches  us  that  the  density  of  water  is 

(8)  p-l_g£am. 


(cent.)^ 

To  find  the  density  of  water  in  terms  of  the  units  inch  and  pound, 
we  use  the  relations 

1  gram  =  0.0022  lb.,  1  centimeter  =  0.3937  in.,  so  that  (1)  becomes 

o-ram  0.0022        lb.         ^ao^i     lb. 


P 


=  1    a'-^-    ^     ^•^""        "'•     =  0.0361 

(cent.)s      (0.3937)3  (in.)^  (in.)^ 


The  ratio  of  the  density  of  any  substance  to  the  density  of 
water  is  called  the  specific  gravity  of  that  substance.  Since 
this  is,  by  definition,  a  ratio  of  two  quantities  of  the  same 
kind,  the  specific  gravity  of  a  substance  is  independent  of 
the  units  of  length  and  mass.  It  is  a  pure  number  and 
therefore  has  no  dimension. 

45.  Velocity.  If  a  train  makes  a  run  of  120  miles  in  4 
hours,  we  say  its  velocity  is  120/4  or  30  miles  per  hour. 
More  generally  ;  if  a  uniformly  moving  body  describes  a 
distance  of  s  length  units  in  t  time  units,  it  would,  at  that 
rate,  describe  s/t  length  units  in  a  single  time  unit,  and  we 
call  the  number 
(1)  ?=z>, 

the  velocity  or  speed  of  the  body.     This  formula  gives  v  =  \ 
when  s  =  1  and  t  =  \.     Therefore  this  definition  of  velocity 


64  LINP:AR   functions   and   progressions     [Art.  46 

includes  also  a  definition  of  a  unit  of  velocity.  The  unit  of 
velocity  is  the  velocity  of  a  body  which  moves  at  the  rate 
of  one  length  unit  per  time  unit.  Therefore  the  unit  of 
velocity  will  change  if  the  unit  of  length  or  the  unit  of  time 
or  both  are  changed.  Since,  in  (1),  s  is  a  length  and  t  a 
time,    the  symbol  for  a  unit  of  velocity  is  L/T. 

Thus  we  may  write 

.,^    mi.        .,^5280  feet      ..,,       5-J8()  feet       ..p. n  feet        , ,  feet 
hour  60  min.  60   min.  min.  sec. 

From  (1)  we  find 
(2)  8  =  vt, 

that  is,  the  distance  described  by  a  uniforndy  moving  body 
varies  as  the  time.  The  factor  of  proportionality  is  v,  the 
velocity  of  the  body. 

46.  Acceleration.  Consider  the  motion  of  a  train  which 
has  not  yet  reached  its  full  velocit}'.  At  a  certain  moment 
let  its  velocity  be  6  feet  per  second ;  and  five  seconds  later 
let  its  speed  be  28  feet  per  second.  Then  the  velocity  of 
the  train  has  increased  22  feet  per  second  in  5  seconds,  or  at 
the  rate  of  4.4  feet  per  second  every  second.  This  is  ex- 
pressed by  saying  that  the  average  acceleration  of  the  train 
in  this  interval  of  time  is  4-4  f^-^^  P^^  second  ])er  second. 

In  general,  if  the  velocity  of  a  body  is  measured  by  v 
velocity  units  at  a  certain  time,  and  by  v'  velocity  units 
after  t  time  units  have  elapsed,  the  velocity  has  changed 
v'  —  V  velocity  units  in  t  time  units,  that  is,  at  the  rate  of 

0)  ^" 

velocity  units  per  time  unit.  This  quotient  is  called  the 
average  acceleration  of  the  body  during  this  interval  of  time. 
If  the  change  in  velocity  is  the  same  for  every  second  of  this 
time  interval,  the  average  acceleration  is  the  same  as  the 
actual  acceleration,  and  is  said  to  be  constant. 

If  a  body  moves  in  such  a  way  that  its  velocity  increases 
by  a  single  velocity  unit  in  a  single  time  unit,  it  is  said  to 


Akt.  47]         UNIFORMLY   ACCELERATED  MOTION  65 

have  unit  acceleration.  In  fact,  the  expression  (1)  reduces 
to  unity  if  v'  —v  is  equal  to  a  unit  of  velocity  and  if  t  is 
equal  to  a  unit  of  time.  If  the  units  of  length  and  time 
are  a  foot  and  a  second,  the  unit  of  acceleration  is  that  of  a 
body  whose  velocity  increases  by  one  foot  per  second  every 
second.  Thus  the  train  in  the  abuve  example  has  an  accel- 
eration of  4.4  acceleration  units. 

Since  a  unit  of  velocity  has  the  dimension  L/T  (Art.  45), 
and,  since  according  to  what  we  have  just  seen,  a  unit  of 
acceleration  is  obtained  by  dividing  a  unit  of  velocity  by  a 
unit  of  time,  the  symbol  for  a  unit  of  acceleration  is  L/T^. 

If  velocities  in  a  certain  direction  are  regarded  as  posi- 
tive, those  in  the  opposite  direction  will  be  regarded  as  nega- 
tive. If,  in  (1),  V  and  v'  are  both  positive  and  if  v'  is 
greater  than  v,  the  acceleration  computed  by  (1)  will  be 
positive.  If  v'  is  less  than  v,  (1)  will  give  a  negative  result. 
Thus,  for  a  positive  velocity  a  negative  acceleration  has  the 
significance  of  a  retardation. 

47.  Uniformly  accelerated  motion.  Let  us  consider  the 
case  of  a  uniformly  accelerated  body,  that  is,  one  whose 
velocity  changes  by  the  same  amount  in  equal  intervals  of 
time.  Let  us  count  time  in  seconds  from  a  certain  instant 
on,  say  6  a.m.,  and  let  the  velocity  of  the  body  be  Vq  feet  per 
second,  tQ  seconds  after  6  A.M.  At  any  other  time,  t  seconds 
after  6  A.M.,  let  the  velocity  be  v  feet  per  second.     Then 

.^.  V  —  Vn      change  in  velocitv 

(1)  a  = -0  = f : r-^ 

t  —  t^  time  elapsed 

will  be  the  average  acceleration,  expressed  in  feet  per  second 
per  second,  during  the  interval  t  —  ^q.  If  the  acceleration 
is  constant,  we  sliall  obtain  the  same  quotient  a  from  (1)  no 
matter  what  value  we  use  for  t,  provided  v  represents  the  veloc- 
ity at  that  instant.     Consequently  we  find  that  the  equation 

(2)  v-v,  =  a(t-t,\ 

obtained  from  (1)  by  clearing  of  fractions,  will  be  true  for 
all  values  of  t  as  long  as  the  acceleration  a  remains  constant. 


66      LINEAR  FUNCTIONS   AND   PROGRESSIONS     [Arts.  48,49 

That  is,  the  change  in  the  velocity  of  a  hody^  which  moves  with 
a  uniformly  accelerated  motion^  varies  as  the  time  which  has 
elapsed.  The  constant  of  variation  in  this  case  is  a,  the  con- 
stant acceleration. 

48.  Falling  bodies.  The  simplest  instance,  in  nature,  of 
uniformly  accelerated  motion  is  that  of  a  falling  body.  The 
constant  acceleration  of  such  a  body  is  usually  denoted  by 
^,  and  simple  experiments,  with  the  Atwood  Machine  for 
instance,  show  that  the  value  of  g  is  approximately 

(1)  ^=32.2      ^^^\  „ 

(second)'' 

if  the  foot  and  the  second  are  used  as  units  of  measurement. 
Let  us  begin  to  count  time  from  the  moment  the  body 
begins  its  motion,  and  let  v^  be  its  initial  velocity.  Then  we 
may  put  ^^  =  0,  a  =  g,  in  equation  (2),  Art.  47,  giving  the 
equation 

(2)  v  =  v^+gt 

for  the  velocity  of  a  falling  body  at  the  erid  of  t  seconds  if  its 
initial  velocity  (velocity  at  the  time  t  =  0}  is  v^  feet  per 
second.  In  this  equation  v  and  Vq  are  positive  for  downward 
and  negative  for  upward  velocities. 

If  the  body  falls  from  rest,  Vq  is  equal  to  zero.  If  it  is 
thrown  downward,  v^  is  positive ;  if  it  is  thrown  upward,  Vq 
is  negative. 

49.  The  importance  of  dimensional  symbols.  —  The  illus- 
trations just  given  will  suffice  to  justify  the  following  re- 
marks. Algebra  strictly  speaking  deals  only  with  abstract 
numbers.  Algebra  may  be  applied  to  the  discussion  of  con- 
crete problems  only  by  introducing  for  every  concrete  quan- 
tity a  certain  quantity  of  the  same  kind  as  unit.  In  dealing 
with  concrete  problems  it  is  very  important  to  specify  what  units 
have  been  employed.      Since  it  is  often   convenient  to   change 

from  one  system  of  units  to  another,  we  must  learn  how  to  make 
such  transformations,  and  this  is  done  most  conveniently  by 
means  of  the  dimensional  symbols. 


Art.  49]     IMPORTANCE  OF   DIMENSIONAL  SYMBOLS  67 

An   equation   between   two   concrete   quantities  is  really  an 

equation  between  the  numerical   measures  of  these  quantities. 

•  Such  an  equation    implies  equality  between  the  corresponding 

concrete  quantities  only  if  they  are  quantities  of  the  same  kind, 

and  if  they  are  measured  in  terms  of  the  same  unit. 

Thus  it  is  obviously  meaningless  to  say  that  40  feet  =  40  square  feet. 
We  have  here  two  concrete  quantities  whose  numerical  measures  are 
equal.  But  this  does  not  imply  equality  of  the  two  concrete  quantities 
themselves,  because  the  latter  are  different  in  kind,  that  is,  have  different 
dimensions. 

We  may  also  express  this  as  follows  :  An  equation  between 
concrete  quantities  has  a  concrete  (^not  merely  a  numerical^ 
significance,  only  if  all  of  the  terms  of  the  equation  have  the 
same  dimension. 

This  fact  often  enables  us  to  find  the  dimension  of  a  quantity  which 
may  otherwise  not  be  apparent,  as  in  the  following  example.  A  familiar 
law  of  Mechanics  (see  Art.  48)  states  that  a  body,  falling  from  rest, 
acquires  after  t  seconds  a  velocity  of  v  feet  per  second,  where 

(1)  (■  =  :32.-2/. 

The  left  member  (a  velocity)  has  the  dimension  L/T  and  the  right 
member  seems  to  have  the  dimension  T.  But  we  have  just  seen  that 
both  members  ought  to  have  the  same  dimension.  The  cause  for  this 
apparent  contradiction  lies  in  thinking  of  32.2  as  an  abstract  number. 
The  equation  itself  tells  us  that  32.2  is  equal  to  v/t,  and  therefore  is 
not  an  abstract  number.  It  has  the  dimension  of  a  velocity  divided  by 
a  time  or  L/T-.     A  more  adequate  \\a,y  of  w-riting  (1)  would  be 

(2)  V  -  yt, 

where  g  is  the  numerical  measure  of  a  quantity  of  the  dimension  L/T^, 
and  where  the  numerical  value  of  g  becomes  equal  to  32.2  only  if  the 
units  of  length  and  time  are  the  foot  and  the  second  respectively.  In 
fact  g  is  the  acceleration  of  a  falling  body  as  explained  in  Art.  48.  The 
dimension  of  g  now  being  known,  we  may  find  the  value  of  g  in  terms 
of  the  units  centimeter  and  minute,  or  in  terms  of  any  other  units 
whatever. 


68  LINEAR   FUNCTIONS   AND   PROGRESSIONS     [Art.  49 

EXERCISE  IX 

1.  Devise  a  vernier  for  making  readings  to  ^^  of  an  inch  on  a  foot- 
rule  which  is  divided  so  as  to  read  directly  to  one-fourth  of  an  inch. 

2.  Show  how  to  construct  a  vernier  which  shall  enable  the  observer 
to  read  angles  to  a  minute  of  arc,  if  the  circle  is  divided  into  half- 
degrees. 

3.  The  area  A  of  a,  circle  varies  as  the  square  of  its  radius  r.  Ex- 
press this  as  an  equation.  What  is  the  value  of  the  factor  of  propor- 
tionality?    Is  it  an  abstract  number?     If  not,  what  is  its  dimension? 

4.  A  glass  beaker  has  a  graduated  scale  upon  it  enabling  an  ob- 
server to  read  oft'  the  volume  of  fluid  which  it  contains.  By  immersing 
spheres  of  various  sizes  in  the  fluid,  it  is  found  experimentally  that  the 
volume  F  of  a  sphere  varies  as  the  cube  of  its  radius,  and  that  V  =  4.19 
cu.  in.  when  r  =  1  in.     Express  F  as  a  function  of  r. 

5.  The  distance  d  (in  miles)  traveled  by  a  train  varies  as  the  time  t 
(in  hours)  counted  from  the  moment  of  its  departure,  and  d  =  150  when 
t  =  5.  Express  this  by  an  equation.  What  is  the  dimension  of  the  con- 
stant of  variation,  and  what  is  its  physical  significance?  How  will  the 
equation  be  modified  if  the  distance  is  expressed  in  feet  and  the  time  in 
minutes? 

6.  The  number  of  feet  s  which  a  body  falls  (from  rest)  in  /  seconds 
varies  as  fi,  and  s  =  64.4  when  t  =  2.  State  this  as  an  equation.  What 
is  the  dimension  of  the  constant  of  variation,  and  how  is  the  equation 
modified  if  s  is  expressed  in  yards  and  t  in  minutes? 

7.  The  volume  of  a  right  circular  cylinder  varies  jointly  as  the  square 
of  the  radius  of  its  base,  and  its  altitude.     Write  this  as  an  equation. 

8.  The  specific  gravity  of  cast  iron  is  7.2.  Find  the  mass  (in 
grams)  of  a  rectangular  block  of  iron  of  dimensions  2  centimeters  x  3 
centimeters  x  4  centimeters. 

9.  A  certain  amount  of  air  is  inclosed  in  a  cylinder  which  has  a 
movable  piston.  The  volume  of  the  inclosed  air  may  be  changed  by 
moving  the  piston.  Show  that  the  density  of  the  inclosed  air  varies 
inversely  as  the  volume. 

10.  The  volume  of  a  sphere  varies  as  the  cube  of  its  radius.  If  three 
spheres  of  radius  2,  3,  and  4  inches  respectively  be  melted  and  formed 
into  a  single  sphere,  what  is  the  radius  of  the  latter?  (Solve  without 
making  use  of  the  actual  value  of  the  constant  of  variation.) 

11.  A  solid  glass  sphere  2  inches  in  diameter  is  melted  and  blown 
into  a  hollow  spherical  shell  whose  outer  diameter  is  4  inches.  Find  the 
thickness  of  the  shell. 


Ai:t.  :)0]  GRAPHrCAL    REPRESENTATION  69 

12.  The  safe  load  of  a  horizontal  beam  supported  at  both  ends  varies 
jointly  as  the  breadth  and  the  square  of  the  depth,  and  inversely  as  the 
length  between  the  supports.  Plan  some  experiments  to  prove  this  Taw 
and  state  it  in  the  form  of  an  equation.  Determine  the  factor  of  pro- 
portionality for  a  certain  kind  of  wood,  if  the  maximum  safe  load  for  a 
beam  15  feet  long,  3  inches  wide,  and  6  inches  deep  is  1800  pounds. 
State  explicitly  what  units  must  be  employed  in  applying  the  residting 
equation.  Find  the  safe  load  of  a  beam  made  of  the  same  material,  18 
feet  long,  4  inches  wide,  and  4  inches  deep. 

13.  The  planets  describe  approximately  circular  orbits  around  the 
Sun.  The  time  required  for  a  planet  to  make  one  revolution  in  its  orbit 
is  called  its  period.  For  the  Earth  this  period  is  one  year.  By  compar- 
ing the  distances  and  periods  of  the  various  planets,  Kepler  *  discovered 
the  following  law,  usually  called  Kepler's  third  law :  in  the  solar  system 
the  square  of  the  period  of  a  planet  is  proportional  to  the  cube  of  its  distance 
from  the  Sun.  Express  this  law  in  the  form  of  an  equation,  and  deter- 
mine the  factor  of  proportionality  if  the  Earth's  distance  from  the  Sun 
be  taken  as  unit  of  distance,  and  one  year  as  unit  of  time.  What  is  the 
distance  of  Jupiter  from  tlie  Sun  if  Jupiter's  period  is  11.86  years? 

14.  At  the  earth's  surface  a  body  falls  193  inches  in  the  first  second. 
The  number  of  inches  which  a  body  not  at  the  earth's  surface,  would 
fall  in  the  first  second  is  inversely  proportional  to  the  square  of  the  dis- 
tance of  that  body  from  the  earth's  center.f  If  the  distance  from  the 
earth's  center  to  the  moon  is  sixty  times  the  earth's  radius,  how  far 
would  a  body  fall  in  the  first  second  if  it  were  as  far  away  as  the  moon? 

50.  Graphical  representation  of  a  pair  of  numbers.  When- 
ever z/  varies  as  a;,  we  have  a  rehition  such  that  to  a  given 
value  of  X  corresponds  a  definite  value  of  i/.     The  numbers, 

*  JoHANN  Kepler  (1571-1(530),  a  famous  German  astronomer.  Kepler's 
greatest  achievement  was  the  discovery  of  the  three  fundamental  laws  of  planet- 
ary motion,  the  la^^t  of  wliich  is  quoted  above.  Tlie  first  of  tliese  laws  asserts  tliat 
a  i>lanet  moves  in  an  ellipse  of  wliirh  the  Sun  occupies  one  locus,  and  the  second 
law  states  that  the  planet's  radius  vector  sweeps  over  equal  areas  in  equal  times. 
Kepler  obtained  these  laws  by  discussing  the  observations  of  the  great  Danish 
astronomer,  Tyiho  BKAnt:.  Newton  showed  later  that  all  these  laws  are  con- 
sequences of  tlie  law  of  gravitation. 

t  This  is  one  way  of  stating  the  law  of  gravitation.  The  calculation  indicated 
in  this  example  was  first  performed  by  Newton.  The  result  of  this  calculation 
did  not  agree  as  well  as  it  shoulil  with  the  observed  motion  of  the  moon  on  ac- 
count of  the  inadequate  notions  curreiU  at  that  time  in  regurd  to  the  dimensions 
of  the  earth.  Newton  therefore  gave  up  the  theory  that  gravitation  varies  ac- 
cording to  the  law  of  inverse  squares  uiUil  a  few  years  later,  wlicii  a  new  survey 
showed  him  a  complete  agreement  with  this  theory.  (See  Grant's  Hhtory  oj 
I'liii.siciif  As/riinn)ni/.  p.  'Jl.) 


M 
Fig.  27 


To  LINEAR   FUNCTIONS   AND   PROGRESSIONS     [Art.  50 

X  and  y,  which  are  so  related  may  be  thought  of  as  belonging 
together  and  constituting  a  pair.  We  shall  now  show  how 
such  a  pair  of  numbers  may  be  represented  geometrically. 

Let  us  draw  two  lines,  unbounded  in  length  and  perpen- 
dicular to  each  other.  We  shall  usually  think  of  one  of  them 
as  horizontal  and  call  it  the  x-axis,  and  call  the  other,  which 
is  vertical,  the  y-axis.  The  point  0,  in  which  the  two  axes 
intersect,  is  called  the  origin  of  coordinates. 

We  adopt  a  unit  of  length,  and  denote  the 
distances  from  any  point  P  to  these  two  axes 
by  X  and  ?/  respectively.     In  Fig.  27,  we  have 

NP=OM=x,  MP=ON=y, 

where  the  notation  is  chosen  in  such  a  way 
that  X  is  measured  on  or  parallel  to  the  a;-axis,  and  y  on  or 
parallel  to  the  ^-axis. 

We  call  X  the  abscissa  and  y  the  ^^ 

ordinate  of  the  point  P.  Both 
numbers  together  are  called  the 
coordinates  of  P.  -j^ 

If  we  take  into  account  only 
the  magnitudes  and  not  the  direc- 
tions of  lines  OM^  ON,  etc.,  that 
is,  if  X  and  y  are  regarded  as 
numbers  without  sign,  there  will 
be  four  points  which  have  the  same  co5rdinates. 

For  instance,  the  points  P,  P',  P",  P'",  in  Fig.  28,  would  all  corre- 
spond to  X  =  3,  y  =  2. 

In  order  to  avoid  this  inconvenience,  we  introduce  the 
convention  that  the  abscissas  of  all  points  to  the  right  of  the 
^-axis  shall  be  positive,  and  of  those  to  the  left  negative  ; 
that  the  ordinates  of  all  points  above  the  a;-axis  shall  be 
positive,  and  of  those  below  negative. 

The  coordinates  of  the  four  points  in  Fig.  28  are  now  different  from 
each  other. 

The  coordinates  of  P  are  x  =+  3,  y  =  +  2, 
The  coordinates  of  P'  are  x  =  —  B,  y  =  +  2, 
The  coordinates  of  P"  are  x  —  —  S,  y  —  —  2, 
The  coordinates  of  P'"  are  x  —  +  3,  y  —  —  2. 


1 


Fig.  28 


Art.  5(1]  GRAPHICAL   REPRESENTATION  71 

The  positive  directions  of  the  x-  and  yaxis,  which  have  now 
been  deiined,  tvill  hereafter  he  indicated  by  a  plus  sign  (as  in 
Fig.  28). 

The  introduction  of  a  coordinate  system  enables  us  to  estab- 
lish a  one-to-one  correspondence  between  the  points  of  the  plane 
(^objects  of  G-eometry^  and  pairs  of  real  numbers  (objects  of 
Arithmetic^.  To  every  loair  of  real  numbers  there  corresponds 
one  and  only  one  point  of  the  plane,  namely  that  one  which  has 
the  given  real  numbers  as  coordinates  ;  and  to  every  point  of  the 
plane  there  corresponds  a  single  pair  of  real  numbers,  namely 
the  coordinates  of  that  point. 

This  method  of  representing  a  pair  of  real  numhers  by  a  point  is,  of 
course,  closely  allied  to  our  former  method  of  representing  a  single  com- 
plex Dumber  by  a  point.  See  Art.  24.*  In  fact  a  single  complex  number, 
z  =  X -{■  yi,  determines  a  pair  of  real  numbers  {x,  y).  It  is  merely  a 
qnestion  of  convenience  whether  we  wish  to  think  of  the  points  of  the 
plane  as  geometrical  representatives  of  a  sbigle  complex  number,  or  of  a 
pair  of  real  numbers. 

We  can  not,  however,  represent  a  joair  of  complex  numbers,  or  even  a  pair 
of  numbers  one  of  which  is  complex,  by  means  of  the  points  of  a  single 
lihine.  If,  therefore,  in  the  solution  of  a  problem  which  involves  the 
coordinates  of  a  point  in  a  plane,  we  find  that  either  of  these  coordinates 
receives  a  comi>lex  value,  we  conclude  that  the  required  point  does  not 
exist.  This  in  no  wise  contradicts  the  fact  that  a  point  of  the  plane  may 
be  represented  by  a  complex  number  x  +  yi,  for,  in  this  symbol  also,  a;  and 
y  are  supposed  to  be  real. 

EXERCISE  X 
Plot  the  points  whose  coordinates  are  given  in  examples  1-4: 

1.  (+3,  +4).  3.    (-:3, -4). 

2.  (-3,   +4).  4.    (+3,   -4). 

5.  If  we  know  nothing  about  a  point  except  that  x  =  0,  what  can  we 
say  about  its  location  ? 

6.  Write  down  an  equation,  involving  one  or  both  of  the  coordinates 
of  a  point  P,  which  will  be  satisfied  provided  that  P  is  anywhere  on  the 
y-axis.  Will  this  equation  be  satisfied  by  any  point  which  is  not  on  the 
y-axis? 

*  We  represented  a  complex  number  x  +  yi  hy  a,  vector,  whose  components 
were  z  and  y.  If  this  vector  is  put  into  its  standard  position,  its  terminus  will  be 
a  point  whose  coordinates  are  x  and  y. 


72 


LINEAR   FUNCTIONS   AND  PROGRESSIONS       [Art.  51 


7,  If  ^/  =  0,  what  can  we  conclude  about  the  position  of  Pi 

8.  Write  down  an  equation  which  will  be  satisfied  it'  and  only  if  P  is 
on  the  X-axis. 

51.  Graphical  representation  of  variation.  If  y  varies  as  x. 
that  is,  a  y  ■=  mx^  there  are  intinitely  many  pairs  of  values,  x 
and  y,  which  satisfy  this  relation.  If  we  plot  a  large  number 
of  these  pairs,  we  find  points  which  seem  to  be  on  a  straight 
line  through  the  origin  of  coordinates. 

^  I  Example.     Let  y  =  2  x.     For  x  =  +  1,    this  gives   y  =  +  2  ; 

for  X  =  +  2,  y  =  +  4 ;  etc.  For  x  =  —  1,  ^  =  —  2 ;  for  x  =  —  2, 
?/  =  —  4  ;  etc.  These  results  are  collected  in  the  adjoining  table. 
If  we  plot  these  pairs  of  numbers  (—3,  —  6),  (—2.  —  4),  etc  , 
we  obtain  the  points  P_3,  P_.^,  P_i,  P^,  P^,  P^,  P^  of  Fig.  29, 
which  seem  to  be  on  a  straight  line  through  the  origin.  The 
idea  naturally  suggests  itself  that  not  merely  the  few  pairs  of 
numbers  which  we  have  calculated,  but  that  all  number  pairs 
which  satisfy  the  equation  ij  =  '2x  will  give  rise  to  points  on 
We  shall  actually />?-ot'e  that  this  is  so  in  Art.  54. 


It  often  happens,  especially  in  concrete 
problems,  that  m  is  so  large  or  so  small  as 
to  render  the  resulting  graph  practically 
useless.  This  may  be  avoided  by  choos- 
+x  ing  different  scale  units  on  the  two  axes. 
These  scale  units  should  always  be  specified 
in  concrete  problems,  so  as  to  make  evident 
the  concrete  significance  of  such  a  diagram. 


-8 

-6 

-2 

-4 

-1 

_  2 

0 

0 

+  1 

+  2 

+  2 

+  4 

+  3 

+  6 

this  line. 


Fig.  29 


EXERCISE  XI 

By  the  method  of  Art.  51  plot  the  graphs  of  the  following  equations: 

1.  y  =  3  X.  4.    y  =  —  x. 

2.  y  —  —  'ix.  5.    2  y  —  3  x  =  0. 

3.  //  =  X.  6.    2  y  +  3  x  =  (». 

7.  Represent  graphically  the  equations  obtained  in  examples  1,  2,  3  of 
Exercise  VIII. 

8.  Represent   graphically   the    equation    obtained    in    example    5   of 
Exercise  IX. 


Arts.  52,  53]        SLOPE   OF    A   STRAIGHT   LINE 


73 


52.  Graphical  representation  of  the  function  y  =  mx  +  b. 
The  metliod  explained  in  the  preceding  article  applies  just 
as  well  to  any  equation  of  the  form 

y  =  mx  +  h. 

We  observe,  in  this  case,  that  the  locus  of  the  points  whose 
coordinates  satisfy  such  an  equation,  again  appears  to  be  a 
straight  line ;  but  this  line  does  not  pass  through  the  origin 
unless  h  is  equal  to  zero. 


EXERCISE  XII 
Draw  the  graphs  of  the  following  equations: 

"i..    y  —  X  -\-  \.  5.    y  =  —  .r  +  1. 

2.   y  =  2x  +  3.  6.   ^  =  -  2.r  +  3. 

Z.   y  =  3x  +  2.  7.   //  =  -  3x  -  2. 

4.   //  =  4x  -  2.  8.   j^  :=  -  4,r  +  2. 

53.  Slope  of  a  straight  line.  Let  Pj  and  F^  be  two  points 
on  a  straight  line,  and  let  the  coordinates  of  these  points  be 
called  (a:j,  y-^  and  (x^^  y^  respectively. 
F'igure  30  shows  that  a  point  P,  in  mov- 
ing from  Pg  to  Pj,  will  move  x^  —  x^  units 
toward  the  right  and  yi  —  y^,  units  upward. 
The  ratio 

(1)  m  =^1  ~  ^2  F^«-^ 

is  called  the  slope  of  the  line. 

Thus,  if  a  railroad  track  rises  3  feet  in  a  horizontal  distance  of 
200  feet,  its  slope  is  =  3/200  =  1.5/100.  It  rises  1.5  feet  in  100  feet;  it 
lias  a  slope  or  grade  of  1.5%- 

If  the  line  P^P^  makes  an  angle  6  with  the  x-axis,  Fig.  30  shows  that 


(2) 


y±ZJb  =  tan  6. 


Consequently,  any  one  who  has  studied  Trigonometry,  may  say  that  the 
slope  ofn  line  in  the  tangent  of  the  angle  which  the  line  makes  tvith  the  x-axis. 

It  is  important  to  note  that  the  definition  of  the  slope,  as 
given  by  (1),  is  independent  of  the  choice  of  the  particular 


74  LINEAR   FUNCTIONS   AND   PROGRESSIONS     [Art.  54 

points,  Pj  and  P^,  provided  both  of  them  are  on  the  line. 
If  we  take  any  two  other  points  of  the  line,  Pg  and  P^,  the 
value  of  m  given  by 

will  be  exactly  the  same  as  that  given  by  (1).  That  this  is 
so  follows  from  familiar  properties  of  "similar  triangles. 
Moreover,  we  might  write,  instead  of  (1), 

(3)  m  =  UiLZLli^ 

interchanging  the  order  of  the  two  points  Pj  and  Pg.     Since 

Vi-  y\  =  -  iy\  -  ^2)'  2^2  -  ^1  =  -  (^1  -  ^2) 

the  values  of  m  given  by  (1)  and  (3)  are  the  same. 

Formula  (1)  may  give  a  positive,  zero,  or  negative  value 
for  the  slope  w  of  a  line.  The  slope  will  be  positive  if  the  line 
rises  from  left  to  right  as  in  Fig.  30.  The  slope  is  negative  if 
the  line  falls  from  left  to  right.  Tlie  slope  is  zero  if  the  line  is 
parallel  to  the  x-axis.  In  that  case  we  have  y,^  =  y^,  and  we 
shall  have 

(^)  y  =  yi 

for  every  point  of  the  line. 

A  line  parallel  to  the  y-axis  cannot  he  said  to  have  any  slope. 
For  such  a  line  we  should  have  x^  =  x^^  so  that  formula  (1) 
which  defines  the  slope  becomes  useless,  since  division  by 
zero  is  excluded  from  Algebra.  (See  Art.  21.)  It  is  clear 
however  that /or  every  poiiit  of  such  a  line.,  the  equation 

(5)  x  =  x^ 

will  he  satisfied. 

54,    Equation  of  a  straight  line.     Let  Pj 

(Fig.  31)  be  a  given  point  and  let  us  con- 
struct the  line  AP  which  passes  through 
Pj,  and  which  has  the  given  number  m  as 
-+a;    its  slope.     If  (a^j,  y^  are  the  codrdinates 
■P^G'  ^^  of  Pj,  we  may  then  regard  2;^,  ?/j,  and  m,  as 


Art.  54]  EQUATION   OF   A   STRAIGHT   LINE  75 

given  numbers.  If  we  denote  by  (.c,  ^)  the  coiJrdiuates  of 
any  other  point  on  AP^  we  must  have 

(1)  .  y^lJh  =  m, 

X  —  T-^ 

since  the  slope  of  AP  miiy  be  computed,  by  formula  (1)  of 
Art.  53,  in  terms  of  the  coordinates  of  any  two  of  its  points. 
Thus,  if  we  regard  x^^  y^,  m  as  given  numbers,  equation  (1) 
will  be  satisfied  by  the  coordinates  (x^  «/)  of  every  point, 
different  from  P^,  of  the  line  AP. 

There  are  no  other  points,  except  those  on  AP^  whose 
coordinates  satisfy  (1).  For,  let  P'  (Fig.  31)  be  any  point 
not  on  AP.  If  we  substitute  iU  coordinates  for  x  and  y  in 
the  left  member  of  (1)  we  shall  obtain,  not  w,  the  slope  of 
AP.,  but  some  other  number  equal  to  the  slope  of  P^P' . 

Consequently,  the  equation  obtained  from  (1)  by  clearing 
of  fractions,  namely 

(2)  y-y\  =  ^(^  -  a^i) 

t's  satisfied  hy  the  coordinates  of  every  point  on  the  line  AP  *  ; 
and  conversely^  every  point  whose  coordinates  satisfy  this  equa- 
tion is  on  AP.     We  express  this  more  briefly  by  saying  that 

(2)  is  the  equation  of  the  line  AP.     Thus,  the  equation  of  a 

line  which  passes  through  the  point  (a:^,  y^  and  ivho><e  slope  is 

m,  is 

y-yi  =  m{x-x^). 

Example.     The  equation  of  the  line  of  slope  2  which  passes  through 

the  point  (3,  4),  is 

jl  —  4  =  J(x  —  .5). 

If  the  given  point  P^  is  on  the  ^/-axis,  coinciding  with  B 
(Fig.  31),  we  shall  have  x-^  =  0,  and  we  shall  usually  denote 
the  ordinate  of  B  by  h.  The  number  b,  whicli  may  be  posi- 
tive, zero,  or  negative  is  called  the  y-intercept  of  the  line. 
Since  we  have  x^  =  0,  ^j  =  6,  equation  (2)  becomes,  in  this 
case, 

(3)  y  =  mx  +  h. 

*  Since  (2)  is  equiv.alent  to  (1)  for  all  points  except  Pi,  and  siuce  the  coordi- 
nates of  Pi  evidently  also  satisfy  (2). 


*y 


76  LINEAR   FUNCTIONS    AND   PROGRESSIONS     [Art.  54 

Therefore,  (3)  is  the  equation  of  the  straiglit  line  whose 
slope  is  771,  and  whose  y-intercept  is  h.  In  otlier  words, 
tlie  graph  of  (o)  will  be  the  straight  line  of  slope  7n  and 
//-intereept  b. 

Tlie  equation  of  a  straight  line  parallel  to  the  ?/-axis  can- 
not be  written  in  either  of  the  forms  (2)  or  (3),  since  the 
slope  of  such  a  line  is  not  defined.  (See  Art.  53.)  It  is 
very  easy,  however,  to  find  the  equation  of  such  a  line.  In 
fact  we  have  shown  already  (Art.  53,  equation  (5)),  that  the 
coordinates  of  every  point  of  such  a  line  satisfy  the  equation 
X  =  x^,  where  x^  is  the  abscissa  of  some  one  point  of  the  line. 
It  is  evident,  conversely,  that  every  point  of 
the  plane,  for  which  x  =  x^.,  is  on  the  line. 

Since  x^  may,  in  particular,  be  regarded  as 
the  abscissa  of  the  point  A  (Fig.  32)  in  which 
the  line  cuts  the  a:-axis,  we  mav  say  that  the 
-^  equation  of  any  line  parallel  to  the  y-axis  may 

Fig.  32  he  written  in  the  form 

(4)  X  =  a-j, 

where  x^  is  the  x-intercept  of  the  line. 

We  have  now  actually  proved  what  the  examples  of  Exer- 
cises XI  and  XII  seemed  to  indicate.  The  graph  of  any 
equation  of  form  (3)  or  (4)  is  a  straight  line.  We  may 
express  this  in  a  little  more  general  form  as  follows.  Tite 
graph  of  any  equation  of  the  form 

(5)  Ax  +  By+C=0, 

where  A,  B,  C  are  any  constants  whatever,  and  tvhere  x  and  y 
occur  only  to  the  first  degree,  is  a  straight  line. 

To  prove  this,  observe  that  equation  (5)  defines  y  as  a 
function  of  x,  namely 

provided  that  B  is  different  from  zero  ;  and  this  function  is 
of  the  form  (3),  so  that  the  graph  of  (6)  is  a  straight  line. 
If,  however,  B  is  equal  to  zero  while  A  does  not   vanish. 


Art.  55]        THE   ZERO   OF    A   LINEAR   FUNCTION  77 

(5)  reduces  to  Ax  +(7=0, 

C 

which  ffives  x  = -• 

^  A 

which  is  of  form  (4),  and  therefore  has  as  its  graph  a  line 
parallel  to  the  ^-axis.  We  need  not  discuss  the  case  A  =  B 
=  0,  since  we  should  then  also  have  (7  =  0,  so  that  all  the 
coefficients  of  (5)  would  vanish.  Such  an  equation  would 
obviously  be  satisfied  by  idl  values  of  r  and  y. 

Smce  the  locus  of  an  equation  of  tlie  form  (5)  is  a  straight 
line,  it  is  unnecessary,  for  the  purpose  of  drawing  the  locus,  to 
calculate  the  coordinates  of  more  than  tivo  of  its  poirits. 

On  account  of  the  fact  that  the  graph  of  the  equation 

y  =  mx  +  l> 

is  a  straight  line,  ever g  function  of  the  form  mx  +  h  is  called  a 
linear  function. 

55.   The  zero  of  a  linear  function.     The  graph  of  the  linear 

function  ,   7 

y  =  mx  +  0 

is  a  straight  line  of  slope  m.  This  line 
will  meet  the  a^-axis  in  a  point  A  unless 
m  happens  to  be  equal  to  zero.  (See  Fig. 
33.)  The  ordinate  of  A  is  equal  to  zero; 
we  wish  to  find  its  abscissa.  Since  A  is 
on  the  line  AB,  the  coordinates  of  A  must  satisfy  the  equation 

(1)  y  =  mx  +  b, 

which  is  satisfied  by  the  cocirdinates  of  every  point  on  AB ; 
and,  since  the  ordinate  of  A  is  zero,  the  abscissa  x  oi  A  must 
satisfy  the  equation 

(2)  0  =  mx  +  b, 

obtained  from  (1')  by  substituting  in  it  y  =  0.  But  (2) 
gives 

(3)  x=-^ 

m 


78  LINEAR   FUNCTIONS   AND   PROGRESSIONS     [Art.  55 

provided  that  m^O.*  If  m  =  0,  equation  (1)  reduces  to 
1/  =  b,  the  corresponding  line  is  parallel  to  the  2;-axis,  and 
there  is  no  point  of  intersection. 

The  particular  value  of  x,  namely/  x  =  —  h/m,  which  causes 
the  function  mx  +  b  to  assume  the  value  zero,  is  called  the  zero 
of  the  function. 

The  zero  of  the  function  mx  +  ^  is  the  same  number  as  the 
solution  or  root  of  the  equation 

mx  +  6  =  0. 

Moreover,  the  zero  of  the  function  mx  +  b  is  equal  to  the 
abscissa  of  the  point  in  which  the  graph  of  the  function  crosses 
the  X-axis. 

EXERCISE  XIII 

1.  Draw  the  lines  for  which  ?h  =  3,  6  =  2 ;  m  =  —  2,  6  =  1 ;  m  —  —  \, 
J  =  —  1 ;  and  write  their  equations.  Find  the  zeros  of  the  corresponding 
functions  both  by  measurement  from  the  figure  and  by  calculation. 

2.  The  equation  .3a:  +  5^  +  4  =  0  defines  i/  as  a  linear  function  of  x. 
Find  this  function  and  its  zero,  draw  the  corresponding  line,  calculate  its 
^/-intercept  and  slope. 

3.  What  is  the  slope  of  a  line  which  joins  the  points  (3,  4)  and 
(5,  7,)  ?     Of  the  line  which  joins  (  -  1,  -  2)  and  (-  4,  3)  ? 

4.  A  line  of  slope  2  passes  through  the  point  (2,  3).  Draw  the  line 
and  find  its  equation  :  What  are  the  .r-  and  ?/-intercepts  of  this  line  ? 

5.  Find  the  zeros  of  the  following  functions : 

3  X  -  4,   7  X  +  9,   8  2-  -  4,  4  X  +  8. 

Solve  the  following  equations  :     (Ex.  9  to  three  decimal  places.) 

6.  8  x  -  5  =  13  -  7  X. 

7.  12  +  3  .r  -  6  -  —  =  —  -  51. 

3         4  * 

8.  ^  +  J  +  ?=.7x-734  +  f. 
2      3     4  5 

9.  3.2.50  .r  -  5.007  -  x  =  0.200  -  0.340  a:. 

10.   -^ ^  =  1.  11.    -^  ^  dc  =  hx-  ac. 

a  —  b      a  +  b  h  —  c 

*  The  symbol  ^  is  read,  is  different  from. 


Art.  55]        THE   ZERO   OF   A   LINEAR   FUNCTION  79 

12.  A  and  B  go  into  partnership.  A  contributes  twice  as  nuich  as  B 
to  their  joint  capital  of  138,700.     What  is  the  contribution  of  each  ? 

13.  A  man  has  f  1.500  in  the  bank  drawing  simple  interest  at  3  %  a 
year.  Express  the  amount  at  the  end  of  n  years  as  a  function  of  n,  and 
represent  this  function  graphically.  What  is  the  financial  significance 
of  the  slope  of  the  resulting  line?  Do  the  ordinates  which  correspond 
to  negative  values  of  n  have  any  significance  ?  What  does  the  ordinate 
mean  which  corresiionds  to  n  =  0? 

14.  (Continuation  of  13.)  A  second  man  has  '11200  in  the  bank, 
drawing  simple  interest  at  4  %  per  annum.  Express  the  amount  of  this 
sum  at  the  end  of  n  years  by  a  formula,  and  represent  this  function 
graphically,  making  use  of  the  same  axes  of  reference  as  in  Ex.  13, 
What  is  the  financial  significance  of  the  point  of  intersection  of  the  twa 
lines  ? 

15.  A  sum  of  money  •*$  P  is  put  out  at  simple  interest  at  the  rate  of 
r%  annually.     Find  a  formula  for  the  amount  A  at  the  end  of  n  years. 

16.  Making  use  of  Ex.  15,  find  a  formula  for  the  sum  of  money  P, 
which  will  yield  an  amount  of  ^^4  at  tlie  end  of  ?i  years,  at  simjile  inter- 
est of  r  %  a  year. 

17.  A  bicyclist  starts  from  a  certain  place  at  8  a.m.  and  rides  at  the 
rate  of  10  miles  per  hour.  An  automobile,  going  with  a  velocity  of  35 
miles  per  hour,  follows  the  same  road,  starting  at  2  p.m.  At  what  time 
will  the  automobile  catch  up  with  the  bicycle?  Solve  this  problem  both 
numerically  and  graphically. 

18.  The  following  abstract  problem  contains  all  examples  like  17  as 
special  cases.  A  starts  from  a  certain  place  and  travels  at  the  rate  of  v 
miles  per  hour.  B  starts  h  hours  later  than  A  and  travels  along  the 
same  road  at  a  rate  of  v'  miles  per  hour.  In  how  many  hours  will  A  and 
B  be  together?  Solve  this  general  problem,  and  discuss  the  solution.  In 
particular,  distinguish  between  the  three  cases  v'  >  v,  v'  =  i%  v'  <  v. 

19.  The  distance  between  two  cities,  A  and  B,  is  180  miles.  An  auto- 
mobile starts  from  A  toward  B  at  8  a.m.  with  a  speed  of  30  miles  per 
hour.  A  second  automobile  starts  from  B  toward  A  at  8  :  30  a.m.  travel- 
ing 35  miles  per  hour.  When  and  where  will  they  meet?  Solve  this 
problem  both  numerically  and  graphically. 

20.  Formulate  and  solve  a  general  problem  of  which  19  shall  be  a 
particular  case.     Discuss  your  solution. 

21.  An  observer  is  stationed  on  a  rowboat  which  is  at  rest.  He 
notices  that  n  wave-crests  pass  the  boat  in  one  second,  and  that  each 
■wave  is  propagated  with  a  velocity  of  Ffeet  per  second.  He  then  starts 
to  row  in  the  direction  of  the  waves  with  a  velocity  of  v  feet  per  second. 


80  LINEAR   FUNCTIONS   AND   PROGRESSIONS     [Art.  56 

How  many  wave-crests  will  pass  the  boat  per  second  while  it  is  in 
motion  ?  Answer  the  same  question  for  the  case  when  the  boat  is  mov- 
ing with  a  velocity  of  v  feet  per  second  against  the  direction  of  the  waves. 

22.  Sound  is  produced  by  waves  or  vibrations  of  the  air,  and  the  pitch 
of  a  tone  depends  upon  the  number  of  vibrations  which  strike  the  ear  in 
one  second.  Making  use  of  Ex.  21,  explain  the  following  phenomenon. 
When  a  train  passes  a  source  of  sound  (a  bell  or  a  whistling  engine), 
the  pitch  of  the  sound  changes.     (Doppler's  principle  for  sound.) 

23.  A  train  with  a  velocity  of  47  feet  per  second  is  passing  a  whistling 
engine  at  rest.  The  velocity  of  sound  is  about  1131  feet  per  second. 
Find  the  ratio  of  the  number  of  sound  vibrations  which  strike  the  ear  of 
an  observer  on  the  moving  train  before  it  reaches  the  engine  to  the  cor- 
responding number  of  vibrations  after  it  has  passed.  (Read  Examples 
21  and  22  to  assist  you  in  solving  this  example.) 

24.  Light  is  produced  by  the  vibrations  of  the  so-called  "ether."  The 
number  of  vibrations  which  strike  the  eye  per  second  will  be  modified  by 
motion  in  the  line  of  sight  either  of  the  observer  or  of  the  source  of  light. 
The  number  of  vibrations  which  strike  the  eye  per  second  may  be  meas- 
ured (l)y  means  of  a  spectroscope).  Explain  how  these  facts  make  it 
possible  to  measure  the  velocity  with  which  a  star  is  approaching  or 
receding  from  the  earth.     (Doppler's  principle  for  light.) 

25.  The  centigrade  thermometer  scale  is  obtained  by  marking  the 
freezing  point  of  water  0°,  the  boiling  point  of  water  100°,  and  dividing 
the  interval  into  100  equal  parts.  On  the  Fahrenheit  scale  the  freezing- 
point  and  the  boiling  point  are  marked  32°  and  212°  respectively.  Ob- 
tain a  formula  for  reducing  temperatures  expressed  in  Fahrenheit  degrees 
to  centigrade  degrees,  and  make  a  graph  of  this  formula. 

56.  Arithmetic  progressions.  If  we  compute  the  values 
of  a  linear  function  7nx  +  b,  not  for  all  values  of  x,  but  only 
for  a;  =  0  and  for  positive  integral  values  of  x,  we  find  a  set 
of  numbers 

(1)  b,  m  ->f  b,    2  w  +  b,    3  m  +  b,    4  m  +  b,  etc. 

each  of  which  differs  from  the  preceding  one  by  the  same 
amount.  These  numbers  form  an  ordered  set  or  a  sequence, 
since  we  are  thinking  of  them  as  being  arranged  in  a  detinite 
order. 

A  finite  set  of  numbers  is  called  an  ordered  set  or  a  sequence^ 
if  the  numbers  of  the  set  are  thought  of  as  being  arranged  in  a 
definite  order. 


Art.  jG]  ARITHMETIC    PROGRESSIONS  81 

If  we  wish  to  discuss  an  ordered  set  of  niiinbers,  it  does  not  suffice  to 
know  the  value  of  every  number  of  the  set.  ^Ve  must  also  know  which 
is  to  be  regarded  as  first,  which  is  the  second,  etc. 

The  numbers  of  an  ordered  set  are  said  to  form  an  arithmetic 
progression,  if  the  difference  hetiveen  any  number  of  the  set, 
after  the  first,  and  the  one  which  precedes  it  is  the  same  for  all 
numbers  of  the  set. 

The  numbers  of  the  set  are  called  the  terms  of  the  pro- 
gression, and  the  difference  between  any  term  and  the  pre- 
ceding one  is  called  the  common  difference. 

The  numbers  (1)  form  an  arithmetic  progression  whose 
first  term  b  is  any  number ;  and  whose  constant  difference 
m  is  any  other  number.  Consequently,  any  arithmetic  pro- 
gression may  be  represented  by  the  formula  (1).  We  find 
the  following  result: 

Tlie  values  which  a  linear  function  mx  +  b  assumes,  when  x 
assumes  in  order  the  values  0,  1,  2,  3,  etc.,  form  an  arithmetic 
progression;  and  converseli/,  with  any  arithmetic  p7'Of/ressioji 
there  may  be  associated  a  linear  function  mx  +  b  whose  values 
coincide  ivith  the  terms  of  the  progression  when  x  is  equated  in 
order  to  0,  1,  2,  3,  etc. 

The  notation  used  in  (1)  is  not  the  one  Avhich  is  usually 
employed  in  the  theory  of  arithmetic  progressions.  Let 
us  call  the  first  term  a  instead  of  b,  the  common  difference 
d  instead  of  m,  and  let  i  denote  the  number  of  any  term,  so 
that  i  replaces  x.     We  then  have  the  customary  expressions 

(2)  a,   a  +  d,    a  -f-  2  (?,    a  -f  3  rZ,  ••• 

for  the  terms  of  an  arithmetic  progression.     The  ith  term 
will  be 

(3)  a+(i-  l)d. 

If  there  are  n  terms  altogether,  and  if  we  denote  the  last 
(wth)  term  by  I,  we  shall  have 

(4)  l=a  +  {n-  l)f?. 


82  LINEAR   FUNCTIONS   AND   PROGRESSIONS     [Art.  56 

The  principal  problem  in  the  theory  of  arithmetic  pro- 
gressions is  that  of  finding  a  formula  for  S,  the  sum  of  all 
of  its  n  terms.     Of  course  we  have 

(5)  >S=a+  (a  +  d)  +  (a  +  -2d)  +  ...  +1. 

But,  in  order  to  compute  the  value  of  jS  by  this  formula,  we 
should  have  to  compute  each  of  the  terms  separately  and 
that  would  be  very  laborious  when  the  number  of  terms  is 
large.  In  order  to  find  a  more  convenient  formula  for  S, 
let  us  first  rewrite  S  by  beginning  with  the  last  term,  that 
is,  by  inverting  the  order  of  the  terms  of  the  progression. 
We  shall  then  have  also 

(6)  S=l+(l-d-)  +  (l-2d)+  ...  +a. 

If  we  add  the  corresponding  members  of  (5)  and  (6)  we 
2S={a  +  l)  +  {a  +  l)  +  {a  +  l)+  ...  +  (a  +  0, 

and  the  right  member  will  contain  the  binomial  a+^  as 
many  times  as  there  are  terms  in  the  progression,  namely  n 
times.     Consequently 

and 

(7)  S=n'^- 

This  formula  is  very  easy  to  remember,  if  we  agree  to  call 
|-(a.  -f  Z),  which  is  half  of  the  sum  of  the  first  and  last  terms, 
the  average  term  of  the  progression.  For,  we  may  then  ex- 
press the  content  of  (7)  as  follows  : 

The  sum  of  an  arithmetic  progression  of  n  terms  is  equal 
to  n  times  its  average  term. 

If  we  substitute  (4)  in  (7),  Ave  find 

(8)  S  =  '^^[2a+in-l)dl 

for  the  value  of  S  in  terms  of  the  first  term,  the  common 
difference,  and  the  number  of  terms. 


Art.  56]  ARITHMETIC   PROGRESSIONS  83 

If  three  of  the  five  quantities  a,  d^  n,  I,  S  are  given,  the 
other  two  may  be  found  from  (4)  and  (7),  or  from  (4)  and 
(8).  It  is  necessary,  however,  to  remember  that  n  must  be 
a  positive  integer,  while  a,  d^  I,  and  S  may  be  positive  or 
negative,  integral  or  fractional,  rational  or  irrational,  real 
or  complex. 

EXERCISE  XIV 

Find  the  last  term  and  the  sum  of  the  following  arithmetical  pro- 
gressions. 

1.  1  +2  +  3  +  •••  +  14. 

2.  2  +  11  +  20  +  •••  to  10  terms. 

3.  -  3  -  8  -  13  -  •••  to  19  terms. 

Compute  I  and  S  in  the  following  cases. 

4.  a  —  2.(1  =  3,  n  =  17. 

5.  o  =  2.5,  d  =  1/3,  n  =  100. 

6.  o  =  1/2,  f/  =  -  1/8,  n  =  20. 

7.  a  =  -10,d  =  -'2,n  =  6. 

In  the  following  examples,  compute  those  of  the  five  quantities,  a,  d, 
I,  n,  S,  which  are  not  given. 

8.  a  =  3,  n  =  333,  S  =  166,833. 

9.  a  =  3/4,  n  =  iO,  S=  517.5. 

10.  d  =  S,n  =  16,  S=  440. 

11.  d  =  2/7,  71  =  32,  S  =  160. 

12.  a  =  1700,  d  =  o,l  =  1870. 

13.  In  Examples  1-7  find  the  sum  when  the  number  of  terms  n  is 
arbitrary. 

14.  Find  a  formula  for  d  when  a,  n,  and  /  are  given. 

15.  Find  a  formula  for  I  when  a,  n,  and  S  are  given. 

16.  Find  a  formula  for  n  when  a,  I,  and  S  are  given. 

17.  Find  a  formula  for  n  when  d,  n,  and  I  are  given. 

18.  Find  a  formula  for  a  when  li,  it,  and  S  are  given. 

19.  Find  the  sum  of  the  first  n  odd  numbers. 

20.  A  ball  rolling  down  a  plane  inclined  at  an  angle  of  30°  to  the 
horizon,  rolls  8  feet  in  the  first  second,  and  in  every  second  thereafter 
it  rolls  16  feet  more  than  in  the  preceding  second.  How  far  will  it  roll 
in  9  seconds? 


8-t      LINEAR   FUNCTIONS   AND   PROGRESSIONS     [Arts.  57, 58 

57.  Insertion  of  arithmetic  means.  To  insert  n  arith- 
metic means  between  two  given  numbers,  a  and  J,  means  to 
find  n  numbers,  a^,  a,^,  a^^  •-•,  a„,  such  that  the  w  +  2  numbers 

(1)  a,  «i,  ^2,  •••,  «„,  h 

shall  form  an  arithmetic  progression.  There  are  n  +  2 
terms  in  this  progression,  a  is  the  first  term  and  b  the  last. 
Hence  we  have  from  (4)  of  Art.  56, 

h  =  a  +  (7i  +  2  —  l}d  =  a  +  (n  +  l)c?, 

where  d  denotes  the  common  difference.  Consequently  we 
find  7 

d=^-'' 


n+  1 
and  therefore 

/-ON  ,   b  —  a  ,   ,-.b  —  a  ,       b  —  a 

^  ^        ^  n  +  1  71+1  n  +  1 

are  the  n  arithmetic  means  required. 
In  particular,  if  n  =  1,  we  find 

a.^^  =  a-\ —  =a-\-l-b—  la=la+^b  =  |-(a  +  i). 

Therefore,  to  insert  a  sinr/le  arithmetic  mean  between  a  and  6, 
ive  need  merely  compute  half  the  sum  of  a  and  b.  This  fact 
accounts  for  the  name  arithmetic  mean  which  is  usually 
given  to  half  the  sum  of  two  numbers. 

EXERCISE  XV 

1.  Insert  six  arithmetic  means  between  3  and  8. 

2.  Insert  five  arithmetic  means  between  3  and  —  2. 

3.  The  two  end  posts  of  a  fence  have  been  placed  at  two  points 
300  feet  apart.  The  fence  is  to  have  35  other  posts.  How  far  apart 
must  they  be  placed  if  all  equally  spaced? 

58.   Harmonic  progressions.     A  sequence  of  numbers  a^,  a^, 

rtg,  etc.  is  said  to  form  a  harmonic  progression,  if  the  recip- 
rocals   of    these    numbers   form  an  arithmetic  progression. 


Art.  59]  HARMONIC   PROGRESSIONS  85 

Consequently  the   general    ex[)ression    for   the   terms    of   a 
harmonic  progression  of  n  terms  is  given  by 

(^         1    1       1 


a    a  +  cC  a  +  '2.d'       '  a+(n—V)d 

The  most  familiar  illustration  of  such  a  progression  arises  if 
in  (1)  we  put  a  =  d=\  ;  this  gives  the  harmonic  progression 

r9>.  1111  ...  1. 

^•^^  1'  2'  3'  4'       '  n 

If  we  wish  to  insert  n  harmonic  means  between  two  given 
numbers,  a  and  6,  we  may  first  insert  ii  arithmetic  means  be- 
tween 1/a  and  1/h.  The  reciprocals  of  these  arithmetic 
means  will  be  the  harmonic  means  between  a  and  h. 


EXERCISE  XVI 

1.  Insert  three  harmonic  means  between  4  and  8. 

2.  Find  an  expression  for  the  harmonic  mean  between  a  and  b. 

3.  Show  that  the  values  of  the  function  which  correspond  to 

mx  +  b 

3:  =  0,  1,  2,  3,  etc.  form  a  harmonic  progression. 

4.  Prove  that  if  all  of  the  terms  of  an  arithmetic  or  harmonic 
progression  are  multiplied  by  the  same  number,  the  result  is  again  a 
progresi^ion  of  the  same  kind. 

59.  Geometric  progressions.  The  ancients  were  familiar 
with  the  essential  properties  of  arithmetic  and  harmonic 
progressions.     They  also  considered  progressions  of  the  form 

(1)  a,  ar,  ar^  ar^^  ar^^   •••, 

in  which,  the  exponents  of  r,  namely  1,  2,  3,  4,  etc.,  are  in 
arithmetic  progression,  and  spoke  of  such  sequences  as 
geometric  progressions.  If  we  divide  any  term  of  the 
sequence  (1),  excepting  the  first,  by  the  term  which  im- 
mediately precedes  it,  we  always  obtain  the  same  quotient, 
namely  r.  We  may  therefore  define  a  geometric  progres- 
sion as  follows: 


86  LINEAR   FUNCTIONS   AND   PROGRESSIONS     [Art.  60 

Let  us  divide  each  term  of  a  sequence  (^excepting  the  first) 
by  the  term  which  immediately  precedes  it.  If  all  of  the  quo- 
tients obtained  in  this  ivay  have  the  same  value  r,  the  sequence  is 
called  a  geometric  progression,  and  r  is  called  the  common  ratio 

of  the  proyression. 

Let  a  be  the  first  term  of  the  geometric  progression  and 
let  r  be  the  common  ratio.  Then  the  various  terms  of  the 
progression  will  be  represented  by  the  expressions  (1). 
The  first  term  is  a,  the  second  is  ar,  the  third  is  ar^,  and  so 
on.  The  ^th  term  is  ar^~^.  If  we  use  the  notation  a.  (not 
to  be  confused  with  the  symbol  a^  for  the  ith  power  of  a),  to 
stand  for  the  ^th  term  of  the  progression,  we  have 

ftj  =  a,  ^2  =  ar,  a^  =  ar\   •••, 

and  in  general,  for  any  value  of  the  integer  i, 

(2)  ai  =  ar'-\ 

If  the  progression  contains  w  terms  altogether,  the  last  or 
nth  term  will  be 

(3)  «„  =  ar"-\ 

Sometimes  the  last  term  is  denoted  by  I,  so  that  we  have 
also 

(4)  I  =  ar'^-\ 

60.   Sum  of   a  geometric  progression  of   n  terms.     Let  us 

denote  by  S^  the  sum  of  a  geometric  progression  of  n  terms, 
so  that 

(1)  Sn=  a  +  ar+  ar^  +  ar^  +  •  •  •  +  ar"*'^. 

If  we  multiply  both  members  of  (1)  by  r,  we  find 

(2)  rSn  =         ar  -{■  ar'^  ■\-  ar-  -\-  •  •  •  +  ar""  ^  +  ar"^. 

If  we  subtract  the  members  of  (2)  from  the  corresponding 
members  of  (1),  we  find 

(3)  ^;  -  r/S;  =  a  -  ar''. 


Art.  61]  GEOMETRIC   PROGRESSIONS  87 

since  all  of  the  other  terms  in  the  right  members  of  (1)  and 

(2)  are  eliniinated  by  this  subtraction.     We  may  write  (3) 

as  follows : 

(l-r)>S'„  =  a(l-r"), 

so  that  we  obtain  finally 

(4)  ^,  =  <1^^ 

1  —  r 

provided  that  r  is  not  equal  to  unity. 

Formula  (4)  enables  us  to  compute  the  sum  S^  of  n  terms  of 
a  geometric  progression  ivhose  first  term  is  a  and  whose  com- 
mon ratio  r  is  different  from  unity. 

If  r  =  1,  the  final  step  in  the  process  used  for  deducing  (4)  is  not  per- 
mitted, since  we  must  never  divide  by  zero,  and  since  1  —  r  would  be 
equal  to  zero  in  that  case.  It  is  very  easy,  however,  to  find  the  formula 
for  S„  in  the  case  when  r  =  1.     We  then  liave 

(5)  S^  =  a  +  a  +  ■••  +  a  —  na. 

The  following  alternative  expressions  for  8^  follow  im- 
mediately from  (4) 

(6)  8^  =  ^ zr-^  = 1-  = ^, 

r— 1  r— 1        r— 1 

where  I  =  ar""^  is  the  last  or  nth  term  of  the  progression. 
We  may  also  write 

^rrx                   a       a  —  ar^  a  ar^ 

(J)  *^n  =  -^ 


-  r        1  —  r      1  —  r 

61.  Geometric  means.  If  three  numbers  a,  5,  c  form  a  geo- 
metric progression,  b  is  said  to  be  a  geometric  mean  of  a 
and  c.     Since  we  shall  then  have 

b^c 
a      b 

we  find  b^  =  ac, 

and  therefore 

(1)  b  =  ±  -JTc. 


88  LINEAR   FUNCTIONS   AND   PROGRESSIONS     [Art.  61 

Thus,  two  positive  numbers  a  mid  c  have  two  geometric  means 
ivhich  are  equal  to  +  Vac  and  —  'Vac  respectively.  Usually 
the  positive  square  root  of  ac  is  called  the  geometric  mean  of  a 
and  c. 

To  insert  n  geometric  means  between  a  and  c  we  must 
find  n  numbers  a^,  a^,  ■■■  «„,  such  that  the  n  +  2  numbers 

shall  form  a  geometric  progression. 

Let  r  denote  the  common  ratio  of  such  a  progression. 
Then  a  will  be  the  first  term,  and  c  will  be  the  (w  +  2)th 
term.  Therefore  we  must  have  (Art.  59,  equation  (3) 
where  n  is  to  be  replaced  by  n  -\-  2), 

c  =  ar""^^, 
whence, 

(2)  .  .=   V„-- 

and  ftj  =  ar,    ag  =  ar^  •  •  •,    a„  =  ar", 

enabling  us  to  compute  all  of  the  desired  geometric  means. 

EXERCISE  XVII 

In  examples  1-4  compute  S^  and  I  from  the  given  quantities. 

1.  a  =  1,  r  =  2,  n  -7.  3.    a  =  8,  r  =  i,  n  =  15. 

2.  a  =  4,  r  =  3,  n  =  10.  4.    a  =  3^,  r  =  f,  n  =  8. 

5.  The  third  term  of   a  geometric  progression  is  3,  and  the  sixth 
term  is  81.     Find  the  tenth  term. 

6.  What  is  the  sum  of  the  first  five  terms  of  a  geometric  progres- 
sion whose  second  term  is  2  and  whose  fourth  term  is  8? 

7.  Insert  one  geometric  mean  between  7  and  252. 

8.  Insert  two  geometric  means  between  2  and  250. 

9.  Find  a  formula  for  I  in  terms  of  a,  r,  and  S^. 

10.  Find  a  formula  for  I  in  terms  of  r,  n,  and  S^. 

11.  Find  a  formula  for  «S„  in  terms  of  r,  n,  and  /. 

12.  Find  a  formula  for  a  in  terms  of  r,  n.  and  /. 

13.  Find  a  formula  for  a  in  terms  of  r,  n,  and  S,^, 

14.  Find  a  formula  for  a  in  terms  of  r,  /,  and  5„. 

15.  Find  a  fornmla  for  r  in  terms  of  a,  I,  and  S„. 


Art.  (52]       INFINITE   GEOMETRIC    PROGRESSIONS  89 

16.  If   a,  n,  and  5„  are  given,  show    that  r  must  be  a  root  of   the 

equation  .,  „ 

'■  S,,         S,  —  a      ^ 

r" -r  +  -^ =  0. 

a  a 

17.  If  n,  I,  and  5„  are  given,  show  that  r  must  be  a  root  of  the  equa- 
tion 


S„  -I  S,-  I 

18.  Prove  the  theorem  :  if  all  of  the  terms  of  a  geometric  progres- 
sion are  multiplied  by  the  same  number,  the  products  also  form  a  geo- 
metric progression . 

19.  Prove  the  theorem  :  if  the  corresponding  terms  of  two  geometric 
progressions  are  multiplied  together,  the  products  also  form  a  geometric 
progression. 

20.  The  story  is  related  that  the  inventor  of  the  game  of  chess  de- 
manded the  following  reward ;  one  grain  of  wheat  on  the  first  field  of 
the  chess  board,  two  on  the  second,  four  on  the  third,  eight  on  the  fourth, 
and  so  on.  How  many  grains  of  wheat  would  he  be  entitled  to  altogether, 
there  being  sixty-four  fields  on  the  chess  board  V 

21.  The  directors  of  a  certain  charity  devised  the  following  plan 
for  raising  money.  They  sent  a  letter  to  one  hundred  people  ask- 
ing each  of  them  to  contribute  one  dollar  and  to  write  to  three  friends 
making  the  same  request  of  them.  The  original  one  hundred  letters 
are  marked  1,  those  sent  out  by  the  original  one  hundred  contributors  are 
marked  2,  and  so  on.  The  chain  is  to  be  broken  when  the  mark  5  has 
been  reached.  If  all  of  the  persons  respond  and  if  there  are  no  dupli- 
cates, how  much  will  the  charity  receive? 

22.  Show  that  the  geometric  mean  of  two  numbers  is  also  the  geo- 
metric mean  of  their  arithmetic  and  harmonic  means. 

62.   Geometric  progressions  with  infinitely  many  terms.* 

Let  us  consider  the  geometric  progression 

(1)  l  +  ^+l+i+-' 

for  which  a  =  1,  and    r  =  |.     According   to  formula  (7)  of 
Art.  60,  the  sum  of  n  terms  of  such  a  progression  is 

C'7\  ,e    _  _J}: (2^"  _  O  _  0/  1  -xn  _  9  _  ^ 

2  2  "" 

*  The  discussion  of  this  subject  may  be  postponed  until  the  chapters  on  limits 
and  series  have  been  reached.  Or  else,  the  theory  of  limits  (See  Chapter  XV) 
may  be  inserted  at  this  place  in  the  course. 


90  LINEAR   FUNCTIONS   AND   PROGRESSIONS     [Art.  62 

If  we  allow  w,  the  number  of  terms,  to  increase  indefinitely, 
the  number  2"~i  will  grow  large  very  rapidly,  and  the  recip- 
rocal of  this  number,  l/2"~^  will  become  very  small  for 
large  values  of  n.  It  can  be  shown  that  l/2'*~i  can  be  made 
as  small  as  may  be  desired  by  merely  choosing  the  integer  n 
large  enough.  We  express  this  by  saying  that  the  fractioji 
l/2"~^  approaches  zero  for  its  limit,  when  n  grows  beyond  all 
hound,  or  in  symbols 

lim  _L  =  0. 

^^00  2"-i 

Consequentl}^  >S'„,  as  given  by  (2),  will  approach  2  as  its 
limit  when  n  grows  beyond  all  bound,  or  in  symbols, 

lim  ^„  =  2. 

We  may  explain  this  situation  by  the  following  geometric 
representation.  In  Fig.  34,  let  the  line  segment  OA  be  two 
units  long,  and  let 

OPi  =  l,  0^2=1  +  1,  OP^  =  l  +  l+\,  0^4  =  1  +  1-^1  +  ^,  etc. 

It  is  clear  that  each  of  these  line-segments  is  obtained  from 
the  preceding  one  by  adding  to  it  just  half  of  what  would  be 

required  to  make  it  exactly  two  units  lonsf. 
'  ""^      Consequently  the  line-segment  obtained  by  a 

large  number  of  such  operations  will  never 
be  exactly  equal  to  two  units ;  but  the  difference  between  2 
and  the  length  of  such  a  segment  may  be  made  as  small  as 
we  please  by  repeating  the  operation  often  enough.  In 
other  w^ords,  the  length  of  the  wth  line-segment  of  this 
sequence,  0P„,  Avill  approach  the  limit  2. 

We  may  argue  in  this  way  whenever  the  common  ratio  r 
is  a  positive  or  negative  proper  fraction.  For  we  shall  prove 
later  (See  Art.  275)  that  the  nth.  power  of  any  proper 
fraction  approaches  the  limit  zero  when  n  grows  beyond 
all  bound,  that  is, 

lim  r"  =  0,  whenever  i  r  I  <  1. 


Art.  0:5]  PERIODIC    DECIMALS  91 

Consequently,  formula  (7)  of  Art.  60,  that  is, 


r 

shows  us  that  S^  will  approach  the  limit 

1  —  r 

when  n  grows  beyond  all  bound,  since  the  second  term  of  (3) 
Avill  have  zero  for  its  limit  if  r  is  a  proper  fraction. 

We  obtain  the  following  theorem  :  If  the  common  ratio  r 
of  a  geometric  progression  is  a  positive  or  negative  proper  frac- 
tion, the  sum  of  the  first  n  terms  of  the  progression  will  approach 
a  finite  limit,  namely 

(4)  S=-^, 

1  —  r  . 

if  the  numher  of  terms  n   is  allowed   to  increase   beyond   all 
bound. 

The  limit  S,  obtained  in  this  way,  is  usually  called  the 
sum  of  the  progression  with  infinitely  many  terms.  There  is 
no  harm  in  using  this  terminology.  But  it  is  necessary  to 
remember  that  this  involves  an  extension  of  the  notion  of  a 
sum,  since  the  original  definition  of  a  sum  was  applicable 
only  to  the  case  of  a  finite  number  of  terms.  Strictly  speak- 
ing, iS  is  not  a  sum  at  all,  but  it  is  the  limit  which  a  certain 
sum  approaches  when  the  number  of  terms  is  increased 
beyond  all  bound. 

63.  Periodic  decimals.  A  periodic  decimal  is  a  geometric 
progression  whose  common  ratio  is  the  reciprocal  of  a  power 
of  ten.     Thus  q         q         ^ 

In  this  case  a  =  3/10  and  r  =  1/10.     Therefore  formula  (4) 
of  Art.  62  gives  3  ^ 

^  10         ^ 


92  LINEAR   FUNCTIONS   AND   PROGRESSIONS     [Art.  63 

furnishing  a  proof  of  the  fact  that  the  decimal  fraction 
.333  •  ••  approaches  the  value  1/3  as  its  limit  as  the  number 
of  places  grows  beyond  all  bound. 

The  value  of  any  repeating  decimal  may  be  found  in  this 
way,  even  if  the  repeating  part  of  the  decimal  is  preceded  by 
several  figures  which  are  not  repeated. 

Thus      1.7414141  •••  =  1.7  +  .041  +  —  +  -^^^  +  ... 

100      (lOO)-^ 

^  17         .041     ^17      .041  ^17       41 
1  -  ik      10       .99       10      990' 

These  examples  illustrate  the  following  general  theorem: 

Every  periodic  decimal  is  equal  to  a  rational  number^  whose 
value  may  he  obtained  by  finding  the  sum  of  a  geometric  pro- 
gression with  infinitely  many  terms. 

Of  course  every  terminating  decimal  is  also  equal  to  a 
rational  number,  that  is,  to  a  quotient  of  two  integers.  In 
the  case  of  a  terminating  decimal,  the  denominator  is  a 
power  of  ten. 

Thus  1.376  is  a  rational  number,  since  it  is  equal  to  1376  -=-  1000. 

Thus,  every  terminating  and  every  periodic  decimal  repre- 
sents a  rational  number. 

The  converse  of  this  theorem  is  also  true. 

That  is,  if  any  rational  number  is  expressed  as  a  decimal., 
this  decimal  must  either  terminate.,  or  else  it  must  be  periodic. 

In  this  statement,  given  without  proof,  it  is  understood,  of  course, 
that  there  may  be  any  number  of  digits  in  a  period,  and  that  the  periodic 
portion  of  the  decimal  may  be  preceded  by  a  finite  number  of  digits 
which  do  not  repeat.  Thus  we  shall  speak  of  327.56431431431  .-.as  a 
periodic  decimal  although  the  first  period  (consisting  of  three  digits)  is 
preceded  by  five  digits  wliich  are  not  repeated. 

Therefore,  if  an  irrational  number  is  expressed  as  a  decimal., 
this  decimal  cannot  be  periodic,  nor  can  it  terminate. 

This  theorem  offers  an  excellent  illustration  of  the  great  power  of  a 
mathematical  argument,  and  its  immense  superiority  over  mere  numerical 


Art.  63]  PERIODIC    DECIMALS  93 

calculation.  Thus  we  have  seen  in  Art.  14  that  a/'J  is  an  irrational 
number,  and  we  have  learned  in  elementary  algebra  how  to  express  v'2 
as  a  decimal.  Xo  amount  of  calculation,  even  if  carried  to  thousands  of 
places,  could  ever  show  us  whether  this  decimal  expression  for  V2  is 
periodic  or  not.  But  our  theorems  enable  us  to  assert  positively,  with- 
out any  calculation  whatever,  that  the  decimal  expression  for  y/2  is  not 
periodic. 

EXERCISE  XVIII 

Find  the  sum  of  the  progressions  with  infinitely  many  terms  given  in 
Examples  1  to  6. 

1-    1  +  ^  +  i  +  i  +  ••••  4-   3  -  1  +  1  -  ^  +  .... 

2.  1  -  1  +  ^  -  I  +  ....  5.    9  +  V  +  ¥  +  If  +  •••• 

3.  ;3  +  1  +  I  +  1  +  ....  6.    40  +  ifa  +  3^6^o  +  .... 

Find  the  values  of  the  following  repeating  decimals  : 

7.  0.1111....  10.    0.8333... 

8.  0.868686.. ..  11.   0.00207900207900.... 

9.  0.142857142857....  12.   0.00854700854700.-.. 

13.  Find   the  sum  of 1 1 h  •••  to  infinity  if  n 

n  +  1       (n  +  1)2       (1  +  n)3  ^ 

is  a  positive  number.     What  can  we  say  about  this  sum  if  n  is  zero? 

If  n  =  - 1  ?    If  n  =  -  2  ? 

14.  A  pendulum  is  brought  to  rest  by  the  resistance  of  the  air,  each 
swing  being  one  tenth  less  than  the  pi-eceding  one.  If  a  certain  point 
on  the  pendulum  describes  a  path  15  inches  long  in  the  first  swing,  what 
will  be  the  total  length  of  the  path  which  it  describes  before  it  finally 
comes  to  rest? 


CHAPTER  III 

QUADRATIC  FUNCTIONS  AND  EQUATIONS 

64.  Standard  form  of  a  quadratic  function.  Any  expression 
of  the  form 

(1)  ax^  4-  Sa;  +  c, 

where  «,  ft,  e  are  constants,  and  where  re  is  a  variable,  is 
called  a  quadratic  function  of  x.  Since  such  a  function 
would  reduce  to  a  linear  function  if  a  were  equal  to  zero,  we 
shall  henceforth  assume  a^O.  Aside  from  this  restriction 
upon  a,  the  coefficients  a,  6,  c  may  be  any  real  or  complex 
numbers,  but  we  shall  usually  confine  our  discussion  to  the 
case  where  a,  ft,  c  are  real.  In  particular,  ft  or  <?,  or  both  ft 
and  c,  may  be  equal  to  zero. 

It  happens  frequently  that  a  quadratic  function  appears  in 
a  more  complicated  form  than  (1),  owing  to  the  fact  that 
various  terms,  which  might  be  united  into  a  single  one,  are 
'written  separately.  When  such  an  expression  is  rewritten 
in  the  form  (1),  we  say  that  it  has  been  reduced  to  its 
standard  form. 

EXERCISE  XIX 

Reduce  the  following  quadratic  functious  of  x  to  the  standard  form, 
and  indicate  the  values  of  a,  h,  and  c  in  each  case. 

1.    5  X  +  4  -  .3  x2.  2.   2  a:^  +  a;  -  .5  X-  -  4  X  +  7. 

3.   mx  +  ix2  +  3  r/x  -  4  +  d.  4.    (mx  +  ky^  +  x-  -  r^. 

65.  Graph  of  a  quadratic  function.  The  method  used  in 
Art.  52  for  making  the  graph  of  a  linear  function  may  also 
be  applied  to  a  quadratic  function.  But  the  graph  will  not 
be  a  straight  line.  The  curved  graph  obtained  by  plotting 
a  quadratic  function  is  called  a  parabola.  The  following 
examples  will  serve  to  make  the  student  familiar  with  the 
form  of  this  curve. 

94 


Art.  65]        GRAPH   OF  A   QUADRATIC   FUNCTION 


95 


% 

li 

-4 

+  16 

-;3 

+  9 

_  2 

+  4 

-  1 

+  1 

0 

0 

+  1 

+  1 

+  2 

+  4 

+  ;3 

+  ;) 

+  4 

+  16 

Fig. 


EXERCISE  XX 

1.  Construct  the  graph  of  the  func- 
tion X-. 

Solution.  We  put  y  =  x-,  and  assign 
to  X  arbitrary  values,  such  as  —  4,  —  3, 
-  2,    -  1,   0,    +  1,    +  2,    +3,    +4,   etc. 

We     compute     the     corre- 
sponding values   of   y   and 

obtain  in  this  way  the  pairs 

of  numbers  indicated  in  the 

table.     Each  of  these  pairs 

of   numbers    we   regard   as 

the  coordinates  of  a  point, 

indicated  in  Fig.   35  by  a 

little     cross.     Finally     we 

join    these     points     by     a 

smooth       curve.      Observe 

that  the  curve  has  a  lowest 
point  or  minimum  at  (0,  0),  and  that  the 
curve   i^  symmetric  with  respect  to  the 

//-axis.     The  point  O  is  called  the  vertex  of  the  parabola,  and  the  axis  of 
symmetry  (the  ^/-axis  in  this  case)  is  called  its  axis. 

2.  Draw  the  graphs  oi  y  =  2  x"^,  y  —  o  ofi,  y  —  \  ^^i  y  =  \  ^'  ^"^^  com- 
pare with  the  graph  oiy  —  x^  as  obtained  in  Ex.  1.  For  the  purposes  of 
this  comparison  it  is  desirable  to  draw  these  five  curves  on  the  same 
sheet,  referred  to  the  same  coordinate  axes. 

3.  Draw  the  graphs  of  y  =  —  x-,  y  —  —  2  x%  y  =  —  'i  x'-,  y  =  —  I  x-, 
y  =  ~  I  x^,  and  compare  with  the  graphs  of  Ex.  2. 

4.  Show  that  the  graph  of  y  =  ax-  may  be  obtained  from  that  of 
y  =  x'^  by  multiplying  all  of  the  ordiuates  of  the  latter  by  a.  Discuss 
the  effect  of  this  multiplication  upon  the  form  and  position  of  the  graph 
according  as  a  is  positive  or  negative,  greater  than  unity,  or  less  than 
unity. 

5.  Draw  the  graphs  of 

y=(x-  1)2,  y  =  (X  -  2)^  y=(x-h  1)-,  //  =  (x  +  2)2, 
and  compare  with  the  graph  of  //  =  x-. 

Hint.  Compute  the  values  of  (x  -  1)'-  etc.,  without  expanding  these 
expressions. 

6.  Show  that  the  graph  of  y  -(x  —  h)'-  may  be  obtained  from  that 
of  y  =  J-  by  moving  the  latter  through  a  distance  of  h  units,  toward  the 
right  or  left  according  as  h  is  positive  or  negative. 


96  QUADRATIC   FUNCTIONS  [Art.  66 

7.  Draw  the  graphs  of 

?/  —  1  =  x%   y  —  2  =  x"^,  //  +  1  =  x^,   1/  +  2  =  x^ 
and  compare  with  the  graph  of  ?/  =  x-. 

8.  Show  that  the  graph  of  //  —  k  =  .r^  may  be  obtained  from  that  of 
1/  =  X-  by  moving  the  latter  through  a  distance  of  k  units,  upward  or 
downward  according  as  k  is  positive  or  negative. 

9.  Show  that  the  graph  of  y  —  k  =  a(x  —  hy  may  be  obtained  from 
that  oi  y  =  x^  by  combining  the  three  operations  indicated  in  Exs.  4,  6, 
and  8.  What  are  the  coordinates  of  its  vertex,  and  what  is  the  equation 
of  its  axis,  these  terms  being  explained  in  ICx.  1? 

10.  Draw  the  graph  of  ?/  =  3  x^  +  6  x  +  .5. 

Hint.  Write  y  —  b  =  3(x'^  +  2x),  and  complete  the  square  in  the 
parenthesis  on  the  right,  giving  y  —  2  =  3(x  +  1)"^.     Then  apply  Ex.  9. 

11.  Draw  the  graph  of  y  =  2  x^  +  i  x  ^  1.  Find  the  coordinates  of 
its  vertex  and  the  equation  of  its  axis. 

12.  Draw  the  graph  of  y  =  —  2  x"^  +  Sx  j-  2.  Find  the  coordinates 
of  its  vertex,  and  the  equation  of  its  axis. 

Hint.     Use  the  method  of  Ex.  10  and  then  apply  Ex.  9. 

66.  The  maximum  or  minimum  of  a  quadratic  function. 
The  parabola,  obtained  in  Ex.  1,  Exercise  XX,  as  the  graph 
oi  y  =  2;2,  has  a  minimum  for  x  =  0,  that  is,  the  ordinate 
which  corresponds  to  a:  =  0  is  smaller  than  that  of  any 
neighboring  point  of  the  curve.  Similarly  the  graph  of 
y  =  —  x\  obtained  in  Ex.  3,  has  a  maximum  for  a;  =  0. 

By  Ex.  9  of  Exercise  XX,  the  graph  of 

(1)  y  -k  =  a(x -  hy 

may  be  obtained  from  that  oi  y  =  x^  by  the  following  opera- 
tions. Firsts  all  of  the  ordinates  of  the  graph  oi  y  =  x^  are 
multiplied  by  a;  secondly,  the  resulting  curve  is  moved  (with- 
out rotation)  through  a  distance  of  h  units  parallel  to  the 
2;-axis  (to  the  right  or  left  according  as  h  is  positive  or  neg- 
ative) and  through  a  distance  of  k  units  parallel  to  the  y-axis 
(up  or  down  according  as  k  is  positive  or  negative).  We 
conclude  that  the  graph  of  (1)  will  have  either  a  minimum 
or  a  maximum  at  the  point  (A,  k)  ;  the  former  if  a  is  positive, 
the  latter  if  a  is  negative.     In  either  case  the  point  (A,  k^  is 


Art.  66]  THE   MAXIMUM   OR   MINIMUM  97 

called  the  vertex  of  the  parabola.  The  axis  of  the  parabola 
passes  through  the  vertex  and  is  parallel  to  the  y-axis.  Its 
equation  will  he  x  =  h. 

In  order  to  prove  that  such  a  maximum  or  minimum  exists 
upon  the  graph  of  awy  quadratic  function 
(2)  1/  =  ax^  +  bx  -\-  e,  a  ^t  0, 

it  will  suffice  to  generalize  the  process  indicated  in  Ex.  10 
of  Exercise  XX.  This  process  really  consists  in  rewriting 
(2)  in  the  form  (1),  that  is,  in  "completing  the  square" 
in  the  trinomial  which  occurs  in  (2).  Thus  we  rewrite  (2) 
as  follows  : 


y  —  a\  x^  -\ —  X  +  -]=  a 


a        4  aV     4  a^      a 


or 

(3)        y  =  a\[x  +  -y-^^ 

which  may  finally  be  written  in  the  form 
J  \2     52  _  4  ^^ 


(4)     y  =  <-  +  2^J 


4a 


Let  us  assume,  in  the  first  place,  that  a  is  positive.  Since 
the  square  of  a  real  number  is  always  positive  or  zero,  (4) 
shows  that  (if  a  is  positive), 

t/  = 1- something  positive, 

for  all  real  values  of  x  except  for  x  =  —  h/2  a,  in  which  case 
we  have  exactly 

(•^)        2:  =  -  --,   1/  = +  0  = 

2a  4a  4a 

Since  1/  is  greater  than  the  value  indicated  in  (5)  for  all 
other  values  of  x,  we  have  the  following  result : 

The  function  y  =  ax'^  +  hx  +  e  possesses  a  minimum  whenever 
a  is  positive.  The  value  of  x  for  u'hich  the  function  assumes 
its  minimum  value  is  —  h/'2  a,  and  the  minimum  value  of  the 
function  is  —  (l)^  —  4  ac)  jA.  a. 


98  QUADRATIC   FUNCTIONS  [Art.  66 

Similarly  it  follows  that  the  funotion  possesses  a  maximum 
whenever  a  is  negative.  The  values  of  x  and  i/  which  corre- 
spond to  the  maximum  are  again  given  by  (5).     The  point 

.r,^                                       h                 5^  —  4  ac 
(6)  -  =  -2^'  2'  = 4^' 

is  called  the  vertex  of  the  parabola  in  either  case,  and  the 
line  for  all  of  whose  points 

(^)  —ta 

is  the  axis  of  the  parabola. 

In  plotting  an  equation  of  the  form  (2),  it  is  advisable  to 
determine  first  the  vertex,  by  means  of  (6),  and  second  the 
axis,  which  passes  through  the  vertex  and  is  parallel  to  the 
y-axis.  The  parabola  will  be  turned  up  or  down  according 
as  a  is  positive  or  negative.  Since  the  parabola  is  sym- 
metric with  respect  to  its  axis,  and  since  we  are  now 
acquainted  with  the  general  appearance  of  the  curve,  we 
may  make  a  fairly  accurate  sketch  of  the  graph  if  we  com- 
pute one  further  point  upon  it.  This  additional  point 
enables  us  to  give  the  parabola  the  proper  spread. 

EXERCISE  XXI 

Draw  the  graphs  of  the  following  quadratic  functions.  Determine 
the  vertex  and  axis  of  each  graph,  and  state  whether  the  vertex  cor- 
responds to  a  maximuni  or  minimum. 

1.  11  =  X-  +  a:  +  1.  3.   y  =  -  x^  +  X  +  1. 

2.  1/  -  3  x-  -  5  X  +  2.  4.   //  =  -  2  x^  -  5  x  +  7. 

5.  Tlie  number  82  is  to  be  divided  into  two  parts  such  that  the 
product  of  these  parts  shall  be  as  great  as  possible  (a  maximum).  Find 
the  parts. 

6.  Generalize  this  problem  (Ex.  .5)  to  the  case  where  the  given  num- 
ber is  any  number  n  instead  of  82. 

7.  Prove  that  the  square  has  a  greater  area  than  any  rectangle  with 
the  same  perimeter. 

8.  Two  straight  railroad  tracks  are  perpendicular  to  each  other  at  a 
point  0.  An  engine  starts  at  a  station  A,  ten  miles  from  O  on  the  first 
track,  and  moves  toward  O  with  a  speed  of  30  miles  per  hour.     A  second 


Akt.  67]     GRAPHIC   DETERMINATION  OF  THE  ZEROS       99 

engine  starts  at  the  same  instant  from  B,  fifteen  miles  away  from  0  on 
the  second  track,  and  moves  toward  O  with  a  speed  of  25  miles  per  hour. 
When  will  the  distance  between  the  two  engines  be  a  minimum  ? 

9.  Generalize  problem  8  for  the  case  where  the  distances  are  OA  =  a, 
OB  =  h,  and  where  the  velocities  of  the  two  engines  are  v  and  v' 
respectively. 

67.  Graphic  determination  of  the  zeros  of  a  quadratic  func- 
tion. It  appears  clearly  from  our  examples,  and  also  from 
the  discussion  of  Art.  66,  that  the  graph  of  a 
quadratic  function 

(1)  1/  =  ax^  -\-  bx  +  c 


+y 


may,  or  may  not,  cross  the  a:-axis.     If  it  does 

cross  the  a;-axis,  in  two  points  A  and  B  (See  ^ig  3G 

Fig.  36),  the  abscissas  of  these  points  are  said 

to  be  zeros  of  the  function ;  because,  if  either  of  these  values 

be  substituted  for  x,  the  corresponding  value  of  the  function 

(1)  will  be  equal  to  zero.     More  specifically,  the  abscissas  of 

A  and  B  are  called  the  real  zeros  of  the  function,  because  they 

are  real  numbers.     Clearly,  then,  the  real  zeros  of  (1)  may 

be  obtained  approximately  by  an  inspection  of  the  graph. 

Obviously  the  quadratic  function  miai?  -\-hx-\-  c),  obtained 
from  (1)  by  multiplication  with  any  non-vanishing  constant 
wi,  has  the  same  zeros  as  (1).  Now  the  zeros  of  the  quad- 
ratic function  ax"^  -^-hx  -\-  c  are  also  called  the  roots  or  solu- 
tions of  the  quadratic  equation  ay?'  +  5a;  +  c  =  0,  which  is 
obtained  by  equating  the  function  to  zero.  Thus,  while 
\kiQ  functioyi  m{ax?' -\- hx -\- c)  is  by  no  means  the  same  func- 
tion as  ay?  +  hx  +  c,  both  functions  have  the  same  zeros,  so 
that  the  tivo  quadratic  equations 

m(a^  -f-  hx  -|-  c)  =  0  and  ay?  -\-  hx  -\-  c  =■  ^ 

are  to  he  regarded  as  equivalent,  inasmuch  as  the  same  values 
of  X  will  satisfy  them  both. 

For  instance,  4(x'^  —  1)  is  a  different  function  from  x^  —  1.  Except 
for  .r  =  ±  I  their  values  are  different,  the  values  of  the  former  function 
being  just  four  times  as  great  as  the  corresponding  values  of  the  hitter. 


100  QUADRATIC    FUNCTIONS  [Art.  68 

But  the  two  functions  are  equal  to  zero  for  the  same  values  of  x,  namely 
for  x=  ±\.  Therefore  the  equations  4(x2— 1)  =  0  and  x^— 1=0  are 
equivalent. 

Since  division  by  a  non-vanishing  number  m  is  equivalent 
to  multiplication  by  the  reciprocal  1/m,  our  remark  justifies 
the  following  statement :  In  solving  a  quadratic  equation^  we 
tnay  inultiply  or  divide  both  members  of  the  equation  by  the  same 
non-vanishing  constant. 

EXERCISE  XXII 

Find  the  real  zeros  of  each  of  the  following  quadratic  functions  by  in- 
spection from  their  graphs,  or  else  show  that  the  function  has  no  real 
zeros : 

1.  x2  -  3  X  +  2.  4.    Zx"^  -X-  4. 

2.  x2  +  6  X  +  5.  5.    x^  +  X  +  1. 

3.  2  x2  -  4  X  +  2.  6.   2  x2  -  4. 

68.    Calculation  of  the  real  zeros  of  a  quadratic  function. 

The  graphic  method  of  determining  the  real  zeros  of  a 
quadratic  function  usually  gives  us  only  approximate  values 
of  these  zeros.  To  determine  their  exact  values  we  return 
to  formulas  (2)  and  (3)  of  Art.  QQ.,  according  to  which 


(1)        y  =  ax^  -\-  bx  +  c  =  a 


b\2      b^-4ac 


If  a,  b,  e  are  real  numbers,  4  a^  is  positive,  but  b^  —  4:  ac  may  be 
positive,  zero,  or  negative.  Let  us  consider  only  those  cases 
at  present  where  6^  —  4a(?  is  positive  or  zero,  but  not  negative. 
Then,  the  square  root  of  P  —  4  ac  is  a,  real  number,  and  we 
may  regard  the  bracket  in  (1)  as  the  difference  between  the 
squares  of  two  real  numbers,  namely. 


y=a 


h   Y     ^V52_4acV 


But  a  difference  between  two  squares  may  be  factored  in 
familiar  fashion,  so  that  we  obtain 


\        'la  la        )\        la  2a       J 


Art.  68]        CALCULATION   OF   THE   REAL   ZEROS  101 

This  expression  shows  that  y  will  become  equal  to  zero  if 
and  only  if  one  of  the  factors  of  the  right  member  is  equal  to 
zero,  that  is,  if  and  only  if  x  is  equal  to 


either r or r . 

la  z  a 

Tlierefore,  the  quadratic  function  ax^  -[-bx  +  c  has  tivo  real 
zeros,  namely 

^o  N                                    —  ^  ±  V62  —  4  ac 
^•'^  "  = 2-a • 

provided  that  b^  —  -i  ac  >  0.  These  two  zeros  coincide  (^become 
identical)  if  6^  —  4  a<?  =  0. 

It  is  easy  to  see  that  there  are  no  real  zeros  if  b^  —  4:  ac  is 
negative.     For  if  b'^  —  4  ac  <  0,  we  have 

_b^  —4:  ac      f. 
4  a^ 

so  that  the  second  term  in  the  bracket  of  (1)  is  positive. 
The  first  term  is  never  negative  for  real  values  of  x.  There- 
fore the  bracket  will  be  positive  for  all  real  values  of  x. 
Consequently  y  will  be  positive  for  all  real  values  of  x  if 
b^—  4  ac  <  0  and  a  >  0.  Similarly  y  will  be  negative  for  all 
real  values  of  a;  if  5^  —  4  ac  <  0  and  a  <  0.  In  neither  case 
will  there  exist  a  real  value  of  x  which  makes  y  equal  to  zero. 
The  values  of  x  which  satisfy  the  equation 

(4)  ar^  +  bx  +  c=0 

are  called  its  roots.  From  this  definition  it  follows  that 
the  zeros  of  the  quadratic  function  ax^  +  bx  -\-  c  are  also  the 
roots  of  the  quadratic  equation  (4),  this  equation  being 
obtained  from  the  function  by  equating  the  function  to  zero. 

Thus  the  roots  of  (4)  are  given  by  (3).     Let  us  denote 
these  roots  by  x^  and  x^,  that  is,  let  us  put 

.r.  _  -  ^  +  V&2  _  4  ag         _  -  ^  _  V^2  _  4  ag 

^'^^         ""1"  2a  '  "^2-  l~a 


102  QUADRATIC   FUNCTIONS  [Art.  69 

Then  we  may  write  (2)  as  follows: 

(6)  y  =  ax^  -\-  hx  -\-  c  =  a(x  —  x-^(x  —  x^. 
Consequently  we  find  the  followmg  relation : 

If  the  quadratic  function  ax^  ■\-  hx  +  c  has  the  factors  x  —  x^ 
and  X  —  x^,  then  the  quadratic  equation  ax^  +  hx  +  c  =  Q  has 
the  roots  x^  and  x^  ;  and  conversely. 

Clearly,  then,  the  problem  of  factoring  the  quadratic 
function  ax^  +  hx  +  c  and  the  problem  of  solving  the  quad- 
ratic equation  a:x?  -\-hx  ■\-  c  =■  ^  are  so  closely  allied  as  to  be 
regarded  as  equivalent.  The  solution  of  either  problem 
implies  that  of  the  other. 

If  we  add  the  two  roots  x^  and  x^  as  given  by  (5),  we  find 

(7)  a^i  +  0^2  =  -  -, 
and  if  we  multiply  these  roots  we  find 

(8)  ^^^^  =  _Ljj2_^52_4^,)j=^. 

That  is  :   the  sum  of  the  two  roots  of  a  quadratic  equation 

ax^  +  hx  -{-  c  =  ^ 

is  equal  to  —  b/a,  and  the  product  of  the  roots  is  equal  to  c/a. 

Thus,  the  product  and  sum  of  the  roots  of  a  quadratic 
equation  may  be  obtained  by  mere  inspection  although  the 
determination  of  the  roots  themselves  requires  the  extraction 
of  a  square  root. 

69.  Another  method  of  deriving  the  formulas  for  the  roots 
of  a  quadratic  equation.  The  formulas  (3)  or  (5),  of  Art. 
68  may  also  be  obtained  as  follows.     If  the  equation 

(1)  ax^  +  bx+  c=  0 

is  given,  we  divide  both  members  by  a  and  transpose  the 
constant  term.     This  gives 

2  ,   ^  <? 

x^  +  -z= 

a  a 


Art.  69]  COMPLETING   THE   SQUARE  103 

We  now  add  6^4  o?  to  both  members  so  as  to  "  complete  the 
square.     We  find 

(2)  TT  ■{ — X -\ = 1 = . 

Finally  we  extract  the  square  root  of  both  members  and  find 

a:  +  -—  =  ±  -—  V5^  —  4  ac, 
2  a  z  a 

whence  finally 

^ON  —  h  ±  Vh^  —  4  ac 

(3)  x  =  — 


z  a 


as  before. 


This  method  is  convenient  to  apply  to  particular  equations  in  practice. 
Theoretically  it  is  inferior  to  the  proof  of  equation  (3)  which  is  given 
in  Art.  68.  The  proof  given  just  now  only  assures  us  that  if  equation 
(1)  has  a  root,  it  will  be  given  by  one  of  the  values  (3).  To  prove 
that  (1)  actually  has  a  root  we  should  take  the  further  step  of  substitut- 
ing each  of  the  values  (3)  for  x  in  the  left  member  of  (1),  and  verifying 
that  the  result  is  actually  equal  to  zero.  This  verification,  which 
is  carried  out  in  Art.  70  for  another  purpose,  is  unnecessary  if  we  prove 
equation  (3)  by  the  method  of  Art.  68. 

EXERCISE  XXIII 

In  examples  1  to  6,  first  factor  the  quadratic  function  by  inspection 
and  then  find  the  roots  of  the  corresponding  quadratic  equation. 

1.  X-  —  S  X  +  2.  4.   2-2  —  (m  +  n)x  +  mn. 

2.  x^  +  3  X  +  2.  5.   3  2-2  -  X  -  4. 

3.  x2  -  X  -  20.  6.    x'^  -  2  X. 

In  Examples  7  to  12,  factor  the  quadratic  function  by  using  the 
formula  (2)  of  Art.  68.  From  the  factored  form  of  the  function,  find 
the  roots  of  the  corresponding  equation  and  observe  that  they  are  the 
same  as  those  given  by  formula  (5)  of  Art.  68. 

7.  x2  +  3  X  +  1.  10.   3  x2  +  4  X  +  1. 

8.  x2  -  3  X  +  1.  11.    2  x2  -  5  X  +  1. 

9.  x2  -  4  X  +  2.  12.   2  x-2  +  6  X  +  3. 


104  QUADRATIC    FCNCTIOXS  [Art.  70 

In  examples  13  to  18,  determine  the  snm  and  product  of  the  roots  by 
inspection. 

13.  3  X-  -f  4  X  -h  4  =  0.  16.    2  jr  -  6  -^  mx-  =  0. 

14.  2  X-  -  4.S  X  -h  1.2  =  0.  17.    -  62  +  6  x  -^  7  x-  =  0. 

15.  11  -  27  X  -  18  z2  =  0.  18.    (1  -  e-)x-  -  2  mx  +  m'  =  0. 

19.  Find  the  most  general  quadratic  function  which  has  x  =  3  and 
X  =  4  as  its  zeros. 

Sdutian.  The  function  (x  -3)(x  -  4)  is  a  quadratic  function  and 
has  the  given  zeros.  If  a  is  any  non-vanishing  constant,  a  (x  —  3)(x  —  4) 
will  stiU  have  the  given  zeros  :  and  this  product  wiQ  still  be  a  quadratic 
function  because  a  is  a  constant,  that  Ls.  does  not  involve  x.  If  a  were 
not  a  constant,  the  product  a(x  -  3)(x  —  4)  would,  in  most  cases,  still 
have  the  required  zeros,  but  it  would  fail  to  be  a  quadratic  function  of  x. 
Thus  a  (x  —  3)  (x  -  4)  is  the  required  function. 

In  examples  20  to  2-5,  find  the  most  general  quadratic  functions  with 
the  given  numbers  as  zeros. 

20.  1.  2. 

21.  -1.-2. 

22.  +  1.  -  2. 

In  Examples  26  to  31,  find  quadratic  equations  whose  roots  are  the 
numbers  indicated. 

26.   2,3.  29.    +3.0. 

30.    —  772.  —  n. 


23. 

2,0. 

24. 

m,  n. 

25. 

m  -{■  n.  m  - 

-  n. 

27.    -  2,  -  3. 


31.        ^  ^ 


28.    -2.-0.  1  -  e      1  -f  e 

70.  Complex  roots  of  a  quadratic  equation.*  We  have 
seen  that  the  equation 

(1)  ajfi  +  hx  +  c  =  () 

has  no  real  roots  if  yo  _  ^  ^^.  ^  q 

The  graph  of  y  =  (1x^^11+0 

in  this  case  does  not  cross  the  ar-axis. 

*  If  Chapter  I  was  omitted,  that  part  of  Chapter  I  beginning  with  Art.  23 
shoald  now  be  discns-sed  nnless  the  student's  high  school  course  covered  imagi- 
nary numbers  suflSciently  well.  If  desired  Art.  70  and  all  subsequent  discussions 
involving  complex  numbers  may  be  omitted. 


Art.  70]  COMPLEX  ROOTS  105 

In  harmony  with  this  fact,  the  values  of  arj  and  x^i  as  given 
by  formulas  (5)  of  Art.  68,  are  not  real.  They  are  com- 
plex numbers,  since  they  involve  the  square  root  of  the  neg- 
ative number  6^  —  4  ac.  But,  although  these  numbers  are 
complex,  they  satisfy  equation  (1).  In  fact,  the  argument 
by  which  they  were  obtained  holds  just  as  well  in  the  case 
b^  —  4  ac  <.  0  as  in  the  case  P  —  4ae  ^  0,  since  the  laws  ac- 
cording to  which  complex  numbers  combine  are  just  the 
same  as  those  for  real  numbers,  excepting  only  the  mono- 
tonic  laws,  and  the  latter  were  not  used  in  deriving  these 
formulae.     (See  Art.  35.) 

We  shall  give  a  direct  verification  of  the  fact  that  the 
expressions  (5)  of  Art.  68  are  roots  of  equation  (1)  whether 
6^  —  4  ae  is  positive,  zero,  or  negative.  In  fact,  this  verifica- 
tion is  valuable  as  a  check  even  for  the  case  b^  —  4  ac  ^  0. 

Let  us,  then,  compute  the  value  of  the  quadratic  function 
ax^  +  bx+  c,  if  we  put  in  it 

_  -  ^  -I-  V^2  _  4  ag 
X  —  x^  — 

la 

We  find 


axj^+  bx^+  c=  a — -^ 


,    7  —  ^  +  V/)^  —  4  ac  , 

+  0 ; h  e 

_  a 


^2b^-4ac-2  b^b^  -  4  ac      -  b^  +  b^b'^  -  4  og      2  ac 
4  a  'la  '2a 

=  J_  [^3  _  2yc  --5V^5=^^5^  -^  +  -&Vt7^-=-4-^  +  2Xc~\  =  0, 

so  that  x^  is  actually  a  solution  of  (1).     The  verification  for 
x^  is  similar. 

Tlius,  while  a  quadratic  equation  may  have  no  real  roots,  it 
always  has  roots,  either  real  or  complex.  Moreover,  we  may 
say  that  it  always  has  tivo  roots,  at  least  if  J^  —  4  ac  is  not 
equal  to  zero.  If  J^  _  4  ac  is  equal  to  zero,  we  see  from  (1) 
Art.  68,  that  ,  .  2 

y  =  ax"^  +  bx  -\-  c  =  alx  +  - —  ] 


106  QUADRATIC  .FUNCTIONS  [Art.  71 

and  may  therefore  be  regarded  as  a  perfect  square.     Thus, 

the  factors  of  ax^  -\-  hx  +  c,  which  are  in  general  distinct,  are 

equal  to  each  other  if 

^>2  _  4  ac  =  0, 


that  is,        ax^  -\-  hx  -\-  c  = 


Va(  x-\-  ~— 
2a 


H^^i:)] 


Hence,  the  quadratic  function  has  two  linear  factors  in 
this  case  also,  even  if  they  are  both  the  same.  Since  the  fac- 
tors of  the  function  are  so  closely  related  to  the  roots  of  the 
equation,  we  shall  say  that  the  corresponding  equation  has 
two  roots,  but  that  these  roots  are  equal  to  each  other  or 
coincide. 

With  this  terminology  now  perfected,  we  have  actually 
proved  the  following  important  theorem  : 

The  quadratic  equation  with  real  coefficients 
(1)  ax?  +  hx  +  c=0 

always  has  two  roots,  namely 


,-,.  -J  +  V62-4a6'  _5_V^>2_4a6' 

(2)         x^=- ,  x^= 

la  la 

These  roots  are  real  and  distinct  if  b^  —  4:  ac  >  0,  they  are  real 
and  coincident  if  h^  —  4:  ac  =  0,  and  they  are  conjugate  complex 
numbers  if  b^  —  4  ac  <i  0.  (See  Art.  34  for  definition  of  con- 
jugate complex  numbers.) 

The  quantity  b^  —  4  ac,  whose  value  enables  us  to  discrim- 
inate between  these  cases,  is  called  the  discriminant  of  the 
quadratic. 

We  may  even  say  that  a  quadratic  equation  (1),  whose  coefficients  a, 
h,  c  are  any  complex  numbers,  always  has  twb  roots  which  are  also  com- 
plex quantities  given  by  equations  (2).  The  algebraic  verification  given 
in  this  article  applies  to  all  such  cases  without  change. 

71.  Various  methods  of  solving  a  quadratic  equation  or  of  fac- 
toring a  quadratic  function.  We  now  know  how  to  solve  any 
(quadratic  equation.      We    may  do    this    by   making    use    of 


Art.  71]  VARIOUS   METHODS   OF   SOLUTION  107 

formulas  (3)  of  Art.  69  ;  or  else  we  may  carry  out  the  pro- 
cess which  was  used  in  deriving  these  formulas  (completing 
the  square  and  extracting  a  square  root);  or  we  may  be  able 
to  factor  the  corresponding  quadratic  function  by  inspection. 
Of  course,  all  of  these  methods  are,  at  bottom,  identical. 
The  student  is  advised,  however,  to  make  use  of  the  second 
method  in  preference  to  the  first,  at  least  until  he  has  mas- 
tered the  processes  involved  so  thoroughly  as  to  make  this 
procedure  unnecessary.  The  great  advantage  of  the  second 
method  is  this  ;  it  involves  all  of  the  essential  principles  arid 
will  not  he  forgotten  if  once  mastered^  whereas  a  formxda^  no 
matter  how  simple  or  convenient^  is  easily  forgotten. 

It  is  often  convenient  to  write  the  quadratic  equation  in 
the  form 
(1)  a3^+'2hx  +  c  =  0, 

denoting  the  coefficient  oi  x  by  2  6  instead  of  h.     The  for- 
mulas   for   the  roots    will   then  become  somewhat   simpler, 
namely 
^n^  _—l>  +  Vb^  —  ac         _  —  ^  —  ^^^  —  <^c 

IZJ  X^ ■,      Xn • 

a  a 

EXERCISE  XXIV 

Solve  the  following  ten  quadratic  equations.  Always  check  your  re- 
sults by  substitution  or  by  computing  the  sum  and  product  of  the  roots 
and  comparing  with  the  values  which  the  sum  and  product  should  have 
according  to  Art.  68. 


6.   X-  +  X  +  1  =  0. 


1.  x2  +  12  X  +  35  =  0.  5.    6  X  -  30  =  3  j;2. 

2.  20,748  -  1616  x  +  21  x^  =  0. 

3.  x2  -  8  X  =  U. 

4.  3  a:2  +  X  =  7.  7.    8  x^  -  7  x  +  34  =  0. 

3  ;-2        01    V.  _  077  SO 

8.  80  X  -I-  ^^  +  ^^ — ^^^^^  =  18.-)9i  -  3  x2. 

4  12 

9.  x2  -  ^-±^x  +  1=0.  10.    (1  -  e^)x-^  -  2  mx  +  m^  =  0. 

ab 

Find  quadratic  equations  whose  roots  are  the  given  complex  numbers. 
11.    1  +  /,  1  -  i.  12.    -h  +  \  J  V3,  -l-i  tV3. 


13.   ^5(7  +  iVl039),  yV(7  -  J  V1039).  14.  1  +  3  i,  1  -  3  i. 


108 


QUADRATIC    FUNCTIONS 


[Art.  71 


Without  solving  the  following  equations,  discuss  the  nature  of  their 
roots,  that  is,  state  whether  they  are  real  and  distinct,  coincident,  or 
iniaginai'y. 

15.  2-2+  llx+  30  =  0.  19.    4x2- 9x  =  5x2-255| -8x 

16.  622  X  =  15  x"-  +  6384.  20.    3  a:2  -  6  x  +  30  =  0. 


17.  x'^  -  X  +  1  =  0. 

18.  3  x2  +  24  X  +  48  =  0. 


21.  18x2  +  24x  +  8  =  0. 

22.  3x--  X  -  4  =  0. 


In  examples  23  to  29,  find  what  value  or  values  k  must  have  in  order 
that  the  quadratic  equation  may  have  its  two  roots  equal. 

23.  x2  +  3  ^•x  +  ^-  +  7  =  0. 

Solution.     In  order  that  the  roots  may  be  equal,  the  discriminant  must 
vanish,  that  is,  we  must  have 

62  -  4  ac  =  9  F  -  4(A:  +  7)  =  0. 

Solving  this  quadratic  equation  for  k  gives  k  =  2  or  —  ^^-. 

Verification.     For  k  =  2,  the  equation  becomes 
x2  +  6x+9=0, 
which  actually  has  two  equal  roots,  each  being  —  3. 

For  k  —  —  4/9,  the  equation  becomes 

x2  -  -V  ^  +  ¥  =  0, 

which  has  two  equal  roots,  each  being  —  7/3. 


24. 


+  A:x  +  4  =  0. 


29. 
30. 


26.    3x2  +  4x  +  ifc  =  0. 
25.   4x2  +  (l  +  A:)x+  1  =  0.  27.   4x2+ ^x  + ^2=0. 

28.    x2(l  +  ni-)  +  2  kmx  +  k^  -  r^  =  0. 
(fi(inx  +  ^")2+  h-x-  —  a%'^. 

Verify  that  1  +  i  is  a  root  of  the  equation  x2  —  2x+  2  =  0,  by 
making  use  of  the  graphic  interpretation  of  the  complex  quantity  given 
in  Art.  24. 

Solution.     The  complex  quantity  x  =  1  +  i  is  represented  in  Fig.  37  by 
the  vector  OP  whose  length  is  equal  to  V2  and  which  makes  an  angle  of 
45°  witli  the  x-axis.     The  square  of  this  vector  may 
be  constructed  by  the  method  of  Art.  31.     It  is  rep- 
resented by   the  vector    OQ  of  length  2  and  am- 
plitude 90°.     Clearly  OP' represents  2x,  and  OP" 
represents   —  2x.     The  parallelogram  construction 
(see    Art.    26)    gives    OP'"   as  representative   of 
x2  —  2  X,    and    OR    represents    the    number    +  2. 
But    the    sum    of   the   vectors    OP"'   and    OR   is 
clearly  equal  to  zero. 
31.    Verify  that  1  —  i  is  a  root  of  the  equation  x2  —  2x  +  2  =  0,  by 
means  of  the  graphic  representation. 


Fig.  37 


Art.  72]  SPECIAL   FORMS   OF   EQUATIONS  109 

32.  Verify  graphically  that  —  ^  +  ^  i  VS  and  —  I  —  \  t  V3  are  roots 
of  the  equation  x"'^  +  x  +  1  =  0. 

33.  If  the  equation  x-  +  ^x  —  14  =  0  has  one  root  equal  to  7,  what  is 
the  other  root  and  the  value  of  k  ? 

Hint.     Use  (7)  and  (8)  of  Art  68. 

34.  Find  the  value  of  k  and  the  second  root  of  x-  +  x  +  8A,-  =  0,  if  one 
root  is  4. 

35.  Find  the  value  of  k  and  the  roots  of  x^  —  x  —  /.•  =  0,  if  the  differ- 
ence between  the  roots  is  equal  to  9. 

36.  What  relation  must  there  be  between  a,  b,  and  c  if  one  root  of 
rtx^  +  6x  +  c  =  0  is  twice  as  great  as  the  other? 

Hint.  Use  (7)  and  (8)  of  Art.  68,  write  Xg  =  2  Xj,  and  eliminate 
x\  and  Xg  between  the  three  equations  obtained  in  this  way. 

72.  Special  forms  of  quadratic  equations.  The  special  cases 
which  arise  if  one  or  more  of  the  coefficients  a,  b,  c  of  the 
quadratic  equation 

ax^  +  bx  +  c  =  0 

are  equal  to  zero,  should  be  mentioned  explicitly. 

1 .  /f  a  =it  0,  6  ^fc  0,  c  =  0,  the  resulting  quadratic,  namely 

ax^  +  6a;  =  0, 

has  one  of  its  roots  equal  to  zero. 

2.  If  a=^0,  b  =  0,  c^O,  ice  obtain  a  so-called  pure  quadratic 

ax^  +  c=  0, 

whose  two  roots  are  numerically  equal  but  opposite  in  sign. 

The  case  a  =  0  cannot  properly  present  itself  here,  since 
the  equation  would  then  cease  to  be  a  quadratic.  We  shall, 
however,  later,  consider  a  case  closely  connected  with  this, 
namely  the  case  where  a  is  regarded  as  a  variable  whose  value 
approaches  zero  as  a  limit. 

3.  //  a  ^  0,  6  =  c  =  0,  the  quadratic  reduces  to 

ax^  =  0, 

both  of  whose  roots  are  equal  to  zero. 

It  is  easy  to  show  that  the  converse  of  each  of  the  three 
statements  1,  2,  3,  is  also  true.  The  proof  is  left  for  the 
student. 


110  QUADRATIC   FUNCTIONS  [Art.  73 

EXERCISE  XXV 

Determine  the  value  or  values  which  k  must  have  in  Examples  1-3  so 
that  the  equations  may  have  one  root  equal  to  zero.  What  will  be  the 
value  of  the  second  root  in  each  case  ? 

1.  8x2-7A•a;  +  2^--  16  =  0. 

2.  2x2  -  5x  +  ^-  -  4  =  0. 

3.  x2  -  kx  +  F  _  4  ^.  +  3  =  0. 

Determine  the  value  or  values  which  k  must  have  in  Examples  4-G  so 
that  the  corresponding  equations  may  have  their  roots  numerically  equal, 
but  opposite  iu  sign.     Solve  the  resulting  equations. 

4.  3x2  -2x+  Tix  -  6  =  0. 

5.  2  kx"^  -  (5  k  +  26)x  +  k^  =  0. 

6.  7  x2  -  (A;2  _  6  X  +  5)x  -  3  =  0. 

What  values  must  k  and  I  have  in  order  that  the  equations  in  Examples 
7-9  may  have  both  roots  equal  to  zero? 

7.  5 x^-16lx  +  kx-  il  +  k  +  Q  =  0. 

8.  3  x2  +  (A-  +  /)x  +  ^•  -  Z  -  1  =  0. 

9.  4  x2  +  (3  k  +  l)x  +  k-  31-2  =  0. 

73.  Equations  of  higher  degree  solvable  by  means  of  quad- 
ratics. It  often  happens  that  an  equation  of  higher  degree 
than  the  second  may  be  solved  by  a  succession  of  quadratic 
equations.  Every  problem  of  geometry,  for  instance,  which 
can  be  solved  by  means  of  ruler  and  compasses  leads  to  such 
equations.*  While  it  is  not  always  easy  to  recognize  equa- 
tions of  this  kind,  the  following  examples  will  furnish  some 
illustrations. 

EXERCISE   XXVI 

1.  Solve  the  equation  x*  -  13  x2  +  36  =  0. 

Solution.  We  may  regard  this  as  a  quadratic  equation  for  x2.  We 
find  x^  =  4  or  9  and  consequently  x  =  ±  2  or  x  =  ±  3.  Each  of  these 
four  values  of  x  satisfies  the  given  equation. 

2.  x"  -  74  x2  =  -  1225. 


I.    (a:  +  iy"  +  4x+*  =  12. 
\  x/  X 


*  In  this  connection  see  Ex.  6,  Exercise  III. 


Art.  74]        RATIONAL  AND  IRRATIONAL   ROOTS  111 

4.  X*  +  2  x8  -  z2  -  2  x  -  3  =  0. 

Hint.     Regard  x^  +  x  =  z  a.s  the  unknown  quantity. 

5.  ax^n  +  ix"  =  c. 

6.  x8  -  8  =  0. 

Hint.     Observe  that  x  =  2  is  one  root  of  this  equation  so  that  x  —  2  is 
a  factor  of  x^  —  8. 

74.  Rational  and  irrational  roots  of  a  quadratic  equation. 

We  have  discussed  carefully  the  character  of  the  roots  of  a 
quadratic  ec^uation  from  the  point  of  view  as  to  whether 
they  are  real  or  complex.  But  there  is  another,  more 
subtle,  distinction  which  is  also  important,  namely  the  dis- 
tinction between  rational  and  irrational  roots. 
Let 

(1)  ax'^-\-bx  +  c  =  0 

be  a  quadratic  equation  whose  coefficients  a,  b,  c  are  ra- 
tional numbers.  (Cf.  Art.  8.)  Each  of  these  coefficients 
may  then  be  expressed  as  a  quotient  of  two  integers,  that 
is,  as  a  fraction.  If  these  three  fractions  be  reduced  to  a 
common  denominator  p^  we  may  write 

I     7      m  n 

a  =  -,  A  =  — ,  c  =  -, 

F         P  P 

so  that  the  equation  (1)  becomes 

-x^  ■] — x+  -=  0. 
p         p        p 

But  this  equation  has  the  same  roots  as  the  equation 

(2)  Ix^  +  mx  4-  w  =  0, 

whose  coefficients  are  integers.     (See  Art.  67.) 
The  roots  of  (2)  are 


^o\  —  m  +  Vw^  —  4ln  —  711  —  Vwz^  —  4  In 

(6)  2*1  =   ■ ,     3*0  =  , 

11  21 

and  they  are  clearly  rational  numbers  if  m^  —  4  /w  is  a  perfect 
square,  so  that  Vw^  —  4  ^Ai  is  an  integer.     If  ni^  —  4  ?n  is  pos- 


112  QUADRATIC    FUNCTIONS  [Art.  75 

itive  and  not  a  perfect  square,  its  square  root  will  be  irra- 
tional. We  may  prove  that  this  statement  is  true  by 
the  method  which  was  used  in  Art.  15  to  show  that  V2  is 
irrational.  From  the  first  equation  of  (8)  we  find,  by  clear- 
ing of  fractions  and  transposing, 


2  Ix-^  -\-  m  =  Vw'^  —  4  In. 

If  x^  were  rational,  the  left  member  would  be  a  rational 
number,  while  the  right  member  is  irrational.  This  is  a 
contradiction,  so  that  a-j  must  be  irrational.  Similarly  it 
follows  that  X2  must  be  irrational.  We  have  proved  the 
following  theorem : 

The  solution  of  a  quadratic  equation  with  rational  coefficients 
may  he  reduced  to  the  solution  of  an  equivalent  equation., 

Ix^  +  mx  -\-  n  =  0 

with  integral  coefficients.  Tlie  roots  of  this  equation  ivill  he 
rational.,  if  and  only  if  the  discriminant  mn?  —  \  In  is  a  perfect 
square. 

EXERCISE  XXVII 

Apply  this  criterion  to  Examples  1  to  8  of  Exercise  XXIV. 

75.   Quadratic  surds.     The  roots  of  the  quadratic  equation 
(2)  of  Art.  74  are  irrational  if 

m^  —  4  Zw  =  c^ 

is  positive  and  not  a  perfect  square.  This  depends  essen- 
tially upon  the  fact  that  the  expressions  (3)  of  Art.  74  for 
these  roots  contain  the  square  root  of  d.  An  irrational  num- 
ber, such  as  V5,  the  square  root  of  an  integer  which  is  posi- 
tive and  not  a  perfect  square,  is  called  a  quadratic  surd.  The 
same  name  is  sometimes  also  applied  to  irrational  numbers  of 
the  form  Vc^  where  d.,  instead  of  being  an  integer,  is  a  ra- 
tional fraction  which  is  not  a  perfect  square.  We  shall  use 
the  term  in  this  more  general  sense. 

The    following   theorem  is   fundamental  in   dealing  with 
such  surds : 


Art.  75]  QUADRATIC    SURDS  113 

//'  a,  b,  a\  h\  and  d  are  rational  numbers^  and  if  d  is  posi- 
tive and  not  a  perfect  square,  so  that  y/d  is  irrational,  then  an 
equatio7i  of  the  form 

(1)  a  +  h^d  =  a' +  h'-^d 
can  subsist  only  if 

(2)  a  =  a'    and   b  =  b'. 
In  fact,  from  (1)  follows 

(3)  a-a'  =  ib'  -  b)^d. 

If  b'  were  not  equal  to  b,  b'  —  b  would  be  different  from  zero 

and  it  would  be  permissible  to  divide  both  members  of  (3) 

by  b'  -  b,  giving 

a  —  a         /  7 

_ =  V  t«. 

b'  -b 

But  this  equation  involves  a  contradiction,  since  one  of  its 
members  is  a  rational  number,  while  the  other  is  irrational. 
Therefore  b'  —b  cannot  be  different  from  zero ;  that  is,  we 
must  have  b'  =  b.  But  this  condition,  together  with  (1), 
shows  that  we  must  also  have  a'  =  a.  Consequently  the 
theorem  is  established. 

We  know  that  irrational  numbers  obey  the  same  funda- 
mental laws  of  addition  and  multiplication  (Laws  I  to  IX 
of  Art.  2)  which  were  originally  observed  to  be  true  for 
positive  integers.  Let  us  then  examine  the  sura,  difference, 
product,  and  quotient  of  two  numbers  of  the  form  a  +  bVd 
and  a'  +  b'Vd,  where  a,  b,  a',  6',  d  are  rational  numbers  and 
where  Vo?  is  irrational.     We  have  immediately 

(a  +  b^d)  +  (a'  +  bWd)  =  a  +  a'  +  (b  +  //)  V^, 
^  ^       (a  +  bVd)  -  (a'  +  bWd)  =  a-a'  +  (b-  b')Vd, 

if  we  make  use  of  the  commutative  and  associative  laws  of 
addition  and  multiplication,  and  also  of  the  distributive  law 
of  multiplication. 

Again  we  find  in  similar  fashion 

(5)  (a  +  bVd)(a'  +  bWd)  =  aa'  +  bb'd  +  i^ab'  +  a'b)Vd, 


114  QUADRATIC   FUNCTIONS  [Art.  75 

and  also,  if  a!  and  h'  are  not  both  equal  to  zero, 

^a\                     a  +  bVd        a  +  h^d       a'  —  b'^d 
(6)  ■ =  = =  X r 

a'+bWd      a'  +  bWd      a'-bWd 

_  aa'  —  bb'd  +  (a'b  —  ab'^-Vd  _  aa'  —  bb'd      a'h  —  aV    /-^ 
~  a'2  _  5/2;^         -         -  a'2  _  5^2^  +  a'^-b'H       ' 

where   the  denominator  a'^  —  b''^d  cannot  be  equal  to  zero. 
For,  if  it  were,  we  should  have 


d='' 


contrary  to  our  assumption  that  d  is  not  a  perfect  square. 

Each  of  the  right  members  in  (4),  (5),  and  (6)  may  be 
rewritten  in  the  form  A  +  B^d,  where  A  and  B  are 
rational  numbers.  We  see,  therefore,  that  the  sum,  differ- 
ence, product,  and  quotient  of  two  numbers  of  the  form  a  +  hVd, 
a'  +  b'  Vd  is  again  a  tiwnber  of  the  same  kind,  the  usual  excep- 
tion, which  excludes  division  by  zero,  being  made  in  the  case  of 
the  quotient. 

This  property  of  the  numbers  a  +  b^d  is  often  expressed 
by  saying  that  they  form  a  field. 

The  process  indicated  in  (6)  is  usually  called  rationalizing 
the  denominator,  and  is  of  great  importance  in  dealing  with 
surds.  The  auxiliary  quantity-  a'  —  b'^d,  used  in  this  pro- 
cess, is  often  called  the  conjugate  of  a'  +b'Vd. 

If  an  expression  involves  more  than  one  quadratic  surd,  it 
may  be  simplified  by  treating  separately  the  several  surds 
which  occur  in  it  by  the  method  here  indicated. 

EXERCISE  XXVIII 

Simplify  the  following  expressions  involving  surds : 

1.  y/2i  +  VM  -VQ. 

2.  2V8 -7v'l8  +  5\/72. 


3.    Vl8a563  +  V50a3?A 

^Ili^.  8. 

V.5  -  1  V3-V2 


7.   ini}/!.  8.    ^  +  ^- 


4. 

(.3. 

4-V5)(2- 

V5). 

5. 

(7 

■f2V6)(9 

-5^/6). 

6. 

(9- 

-7VI3)(^ 

.  -  6V13). 

q 

2 

1 

10          '^ 

+  V3 

V8- 

2 

Art.  76]  SQUARE   ROOT   OF   a  +  bVd  115 

76.*     The  square  root  of  an  expression  of  the  form  a  +  by/d. 

Let  us  again  consider  a  niuuber  of  the  form  a  +  hVd  where  a,  b,  and  d 
are  rational,  but  where  d  is  a  positive  rational  number  which  is  not  a 
perfect  square.  The  square  root  of  a  +  bVd  will  not,  in  general,  be 
expressible  as  a  sum  of  two  quadratic  surds.  There  are  some  cases,  how- 
ever, in  which  this  may  be  done,  and  we  propose  to  answer  the  question 
as  to  what  cases  these  are. 

Let  us  suppose  that  there  exist  two  positive  rational  numbers,  z  and 
y,  such  that 

(1)  Va  +  hVd  =  \/x±  Vy. 

To  avoid  ambiguity  we  shall  assume  here,  as  elsewhere  in  this  book, 
that  the  symbol  y/k  stands  for  the  positive  square  root  of  k  whenever  k 
itself  is  a  positive  number.  We  shall  assume,  moreover,  that  a  +  bVd  is 
a  positive  number,  so  that  it  has  a  positive  square  root.  This  assump- 
tion does  not  prevent  one  of  the  rational  numbers,  a  or  b,  ivom  being 
negative  ;  but  they  may  not  both  be  negative.  One  of  the  two  terms  in 
the  right  member  of  (1)  maybe  negative,  but  not  both.  Since  the  right 
member  as  a  whole  must  be  positive,  equation  (1)  implies  that,  if  there 
is  a  minus  sign  at  all  in  the  right  member,  the  notation  has  been  so 
chosen  that,  of  the  two  positive  numbers  z  and  y,  the  greater  has  been 
called  X. 

If  (1)  hold.s,  we  must  have 

(2)  a  +  bVd  =(y/x  ±\/]/y^  =  X +  y  ±2Vxy. 

Since  a,  b,  x,  and  y  are  rational,  while  y/d  is  not,  the  quantity  Vxy  cannot 
be  rational.  For  if  it  were,  a  +  bVd  would,  according  to  (2),  be  rational. 
According  to  Art.  75  we  therefore  conclude,  from  (2), 

(3)  X  +  y  =  a,    ±2 Vxy  =  bVd, 
whence 

(4)  x  +  y  =  a,   xy  =  I  b^d. 

We  can  easily  form  a  quadratic  equation  of  which  x  and  y  shall  be  the 
roots,  namely  (see  Art.  68  and  Exercise  XXIII,  Ex.  19), 

(z-x)(z-y)=0, 
^'^  z'  -  (x  4-  y)z  +  xy  =  0, 

which  becomes,  on  account  of  (4), 
(.5)  z^  -  az  +  \  b-d  =  0. 

If  this  quadratic  has  irrational  roots,  equation  (1)  will  be  impossible, 
since  we  assumed  x  and  y,  which  are  the  roots  of  (5),  to  be  rational. 
Now  the  discriminant  of  (5)  is  a-  -  b'-d  and  the  coefficients  of  (5)  are 

♦Article  76  and  Exercise  XXIX  may  be  omitted  without  destroying  the  conti- 

uuity. 


116  QUADRATIC   FUNCTIONS  [Art.  76 

rational.     Therefore  (see  Art.  74)  the  roots  of  (5)  are  irrational  unless 
fl2  —  Ifid  is  the  square  of  a  rational  number. 

Thus,  if  ifi  —  }P-d  is  not  a  perfect  square,  (1)  is  impossible.     If  cfi  —  IM 
is  a  perfect  square  both  of  the  roots  of  (5),  namely 


0+ Va2-62rf            a-^/a^  -  hM 
x  = ^ ,   y  = 2 , 

will  be  rational.  Moreover,  if  a  is  positive,  both  x  and  y  will  be  positive. 
This  is  evident  as  far  as  x  is  concerned;  y  will  be  positive  in  this  case 
because  Va-  —  h'^d  is  less  than  Va^  =  a.  If  a  is  negative,  y  will  be  nega- 
tive, so  that  this  case  (o  <  0)  is  excluded,  since  we  have  assumed  that 
both  X  and  y  shall  be  positive. 

If  we  have  a  >  0  and  a-  —  V^d  a  perfect  square,  x  and  y  will  be  positive 
rational  numbers  which  satisfy  equations  (4).  Both  Vx  and  ^y  will 
then  be  real  positive  numbers,  and  we  may  choose  the  sign  +  or  —  in 
(8)  according  as  h  is  positive  or  negative.  After  this  choice  of  sign  has 
been  made,  equations  (3)  will  be  satisfied,  and  we  shall  actually  have 
the  positive  square  root  of  a  +  h^/d  expressed  in  the  form  (1),  with  the 
+  or  —  sign  according  as  h  is  positive  or  negative. 

Our  complete  result  may  be  summarized  as  follows: 

Let  a,  b,  d,  x,  and  y  be  positive  rational  numbers,  such  that  y/d  is  irrational. 
It  is  possible  to  write 


y^ 


(6)  Vn  +  bVd  =  Vx  +  \^y,    Va  -  by/d  =  y/x  -\/, 

if  and  only  if  a?  —  bM  is  a  perfect  square.     The  values  of  x  and  y  will  then 
be  given  by  the  expressions 

a  +  y/cfi  -  bM  a  -  Va^  -  ¥d 

(0  ^  = 7i »  y^ o 


EXERCISE  XXIX 

1.  Examine  the  possibility  of  expressing  v  3  +  2V2  in  the  form 
y/x  -\-  yfif. 

Solution.     In   this   case  a  =  3,   b  =  2,  d  —  2,   so  that   o-  —  b-d  =  1,   a 

perfect  square.     Moreover,  a,  b,  and  d  are  positive.      Therefore  we   find 

from  (7) 

3  +  1      o  '^  -  1      1 

X  =  — -—  =  2,     //  =  -— —  ^  1, 


so  that         V3  +  2  V2  =  V2  +  Vl  :=  1  +  V2. 

Of  course  the  solution  may  also  be  obtained  without  using  the 
formulic  (7),  by  applying  to  this  particular  case  the  process  by  means  of 
which  these  formulas  were  derived.  The  student  should  do  this  to  help 
him  understand  the  general  process. 


Art.  77]     APPLICATION   OF   THE   MONOTONIC   LAWS        117 


2.  Examine   the  possibility  of   expressing  V 4  +  2\/2   in   the   form 
Vj;  +  Vy. 

Solution.     In  this  case  a^  —  b^d  =  16  —  8  =  8,  which  is  not  a  perfect 

square.     Therefore  such  an  expression  is  impossible  for  V 4  +  2V2. 

Examine   in   the  same  way  the   following  numbers,  and   find   their 
square  roots  in  the  form  y/x  ±  Vy  whenever  possible. 

3.  7  +  4V3.  6.    87  -  r2V42. 

4.  5  -  \/24.  7.    I  +  V2. 


5.   28  +  5v'12.  8.    2  -  V4  -  4  a^. 

77.*  Application  of  the  monotonic  laws  of  Algebra  in  nu- 
merical calculations  involving  quadratic  surds.  The  student  has 
learned  in  his  first  course  in  Algebra  how  to  calculate  the  value  of  a 
quadratic  surd,  that  is,  of  a  square  root,  to  as  many  decimal  places  as 
may  be  desired.  Let  x  be  the  exact  value  of  the  surd  (  V2  for  instance), 
and  let  x„  be  the  approximate  value  found  for  it  by  carrying  out  the 
process  of  extracting  the  square  root  to  n  decimal  places.  Then  x„  will 
be  a  decimal  fraction  with  n  digits  to  the  right  of  the  decimal  point,  and 
we  shall  have  x„<x.  If  we  raise  tlie  last  digit  of  x„  by  a  single  unit  and 
call  the  resulting  decimal  fraction  x,/,  then  x„'  will  be  greater  than  x,  so 
that 

(1)  x„<x<x„'. 

Since  the  difference  between  a:„'  and  ar„  is  equal  to  just  one  unit  of  the 
?ith  decimal  place,  that  is,  to  1/10",  and  since  n  may  be  taken  arbitrarily 
great,  we  may  make  the  differences  x  —  x^  and  x„'  —  x  as  small  as  we 
please  by  choosing  n  large  enough. 

In  the  same  way,  a  second  surd  y  may  be  inclosed  between  two 
decimal  fractions,  y^  and  y„',  differing  from  each  other  by  1/10",  so  that 

(2)  //„  <  .V  <  !/n- 

From  the  monotonic  law  of  addition  we  can  now  conclude  that 

(3)  Xn  +  //„  <:x  +  y<  x,/  f  y,/- 

Since  x„'  differs  from  x„  by  1/10",  and  y,,'  differs  from  y„  by  1/10",  x„'  +  y„' 
differs  from  x„  +  //„  by  2/10",  and  x  +  //,  according  to  (3),  can  differ  from 
either  x„  +  y„  or  x,/  +  //„'  at  most  by  something  less  than  2/10".  But  this 
can  be  made  as  small  as  we  please  by  choosing  «  large  enough.  Thus  we 
conclude  that  the  approximate  value  of  a  sum  nf  two  surds  may  be  found 
with  any  desired  degree  of  precision,  by  using  approximate  values  for  the 
surds  themselves  which  are  sufficiently  close  approximations. 

*  Article  77  may  be  omitted  without  destroying  the  continuity. 


118  QUADRATIC   FUNCTIONS  [Art.  78 

Corresponding  theorems  can  be  established  for  the  difference,  product, 
or  quotient  of  two  surds,  and  it  is  in  this  way  that  the  methods,  ordi- 
narily used  for  calculating  with  surds,  are  justified  theoretically. 

Exactly  similar  theorems  hold,  not  only  for  qiiadratic  surds,  but  also 
for  irrational  numbers  of  any  kind.  To  prove  this  it  is  only  necessary 
to  show  that  for  any  irrational  number  x,  decimal  fractions  a;„  and  x„' 
exist,  with  n  digits  to  the  riglit  of  the  decimal  point,  such  that 

(4)  x,/  -  x„  =  — ,     x„  <  X  <  x„'. 

This  may  be  shown  as  follows.  The  irrational  number  may  be  repre- 
sented by  a  certain  line-segment  on  the  x-axis.  (See  Art.  15.)  Apply 
to  it  a  segment  of  unit  length.  Suppose  that  this  unit  segment  is  con- 
tained in  X,  i  times,  and  let  r  be  the  remainder.  Then  i  is  the  integral 
part  of  X,  and  r  is  the^ fractional  part.  Apply  to  r  a  line-segment  1/10  of 
a  unit  long ;  let  it  be  contained  a  times  in  r,  and  let  r'  be  the  remainder. 
Then  . 

X  —  I  -\ ■  +  ?•',     r'<— . 

10  ^10 

Apply  to  r'  a  line-segment  xo^  of  a  unit  long.  Suppose  it  is  contained  h 
times  in  r'  and  let  r"  be  the  remainder.     Then 

10      100  102 

Proceeding  in  this  way  we  may  find  a  decimal  number  a:„,  smaller  than 
X,  but  differing  from  x  by  less  than  1/10"  where  n  may  be  made  as  great 
as  we  please.  The  remainder  will  never  be  zero  if  x  is  irrational,  since 
evei-y  terminating  decimal  is  a  rational  number.  If  we  raise  the  last 
digit  of  x„  by  one  unit,  the  resulting  number  x,/  will  be  greater  than  x, 
and  we  shall  have  found  the  two  numbers  x„  and  x,/  mentioned  in  (4). 

78.  Interpretation  of  negative,  fractional,  and  complex  roots 
in  concrete  problems.  Many  concrete  problems  lead  to  quad- 
ratic functions  and  quadratic  equations.  In  every  concrete 
problem  we  know  beforehand  what  kind  of  things  we  are 
talking  about,  so  that  we  can  state  beforehand  whether  the 
unknown  quantity  x  whose  value  we  are  seeking  ought,  in 
the  nature  of  things,  to  come  out  as  an  integer,  or  whether 
fractional,  negative,  or  even  imaginary  values  may  also  be 
admissible.  Thus,  if  x  represents  a  sum  of  money,  expressed 
in  dollars,  x  may  be  positive  or  negative  according  as  the 
transaction  considered  involves  credit  or  debit,  profit  or  loss. 
If  a;  represents  the  number  of  yards  in  a  piece  of  cloth,  nega- 


AuT.  78]  INTERPRKTATION   OF   ROOTS  119 

tive  values  of  x  would  be  excluded.  But  if  a:  is  a  symbol 
which  stands  for  a  vector  (see  Art.  23),  complex  values  of  x 
are  just  as  admissible  as  real  values. 

A  quadratic  equation  always  has  two  roots.  If  the  equa- 
tion has  been  obtained  as  a  result  of  the  mathematical  formu- 
lation of  a  concrete  problem,  the  question  always  arises  :  Do 
both  of  the  roots  of  this  equation,  which  may  be  positive, 
zero,  or  negative,  real  or  complex,  actually  represent  solu- 
tions of  our  concrete  problem,  or  does  only  one  of  them 
represent  such  a  solution,  or  finally  does  neither  of  them 
give  a  solution  of  the  concrete  problem  ?  This  question  can 
only  be  decided  by  a  discussion  of  the  nature  of  the  concrete 
problem  which  is  under  consideration.  Our  common  sense 
will  tell  us  whether  a  negative,  or  a  fractional,  or  a  complex 
value  of  X  has  any  significance  in  such  a  concrete  problem  ; 
whether  more  than  one  value  is  admissible  ;  and  if  not, 
which  of  the  two  roots  is  the  correct  solution  of  the  problem. 
Finally,  if  neither  of  the  two  roots  should  turn  out  to  be 
admissible,  the  problem  has  no  solution. 

EXERCISE  XXX 

1.  The  product  of  tAvo  consecutive  numbers  is  156.    Find  the  numbers. 

2.  Divide  21  into  two  parts  whose  product  is  108. 

3.  In  an  arithmetic  progression  the  first  term  a,  the  common  differ- 
ence </,  and  the  sum  S  are  given.  Find  a  formula  for  the  number  of 
terms  /;.  Is  the  resulting  formula  actually  applicable  when  the  values  of 
«,  (/,  and  5  are  given  as  arbitrary  numbers?  If  not.  what  conditions 
must  these  numbers  satisfy? 

4.  One  side  of  a  rectangular  garden  is  seven  yards  longer  than  the 
other,  and  the  area  of  the  garden  is  60  square  yards.  AVhat  are  the  di- 
mensions of  the  garden  ?  Are  there  two  solutions  of  this  problem  ?  State 
the  reason  for  your  rejily. 

5.  Given  a  rectangle  of  dimensions  6x8.  A  second  rectangle  is  to 
be  drawn  inside  of  the  first,  having  half  the  area  of  the  outer  one,  and  in 
such  a  way  that  its  sides  shall  all  be  everywhere  at  the  same  distance 
from  the  sides  of  the  outer  rectangle.  Find  this  distance.  How  many 
solutions  are  there,  and  wliy? 

6.  Solve  the  general  problem  obtained  from  Ex.  5  if  the  given  rec- 
tangle has  the  dimensions  a  x  h. 


120  QUADRATIC   FUNCTIONS  [Art.  79 

7.  A  rectangular  mirror  of  dimensions  a  x  b  is  to  be  framed  in  such 
a  way  that  the  area  of  the  frame  shall  be  equal  to  the  area  of  the  mirror,  and 
that  the  outer  perimeter  of  the  frame  shall  be  a  rectangle  similar  to  the 
unframed  mirror.     Find  formula  for  the  width  of  the  molding. 

79.    Uniform   motion   along  a  straight   line.     One  of   the 

most  important  applications  of  linear  and  quadratic  func- 
tions is  concerned  with  motion.  We  have  already  solved 
some  problems  involving  motion  in  Exercise  XIII,  but  we 
shall  now  discuss  the  questions  involved  more  fully. 

Let  us  think  of  the  point  P  in  Fig.  38  as  being  in  motion 
along  the  line  AC  and  let  its  velocity  v  be  constant.     The 
point  is  then  said  to  be  in  uniform  motion 
*^        ^y^  along  a  straight  line.     We  count  time  in 

seconds  from  some  convenient  moment  on, 
(say  6  A.M.).     At  the  time  ^q  (that  is  t^ 


N 


AO 


Ml — M  '^'^    seconds  after  6  A.M.)  let  the  moving  point 

be  at  Pm  and  let  t  denote  the  time  when 


Fig.  38 


0' 


the  point  has  reached  the  position  P. 
Then,  the  time  which  has  elapsed  between  these  two  instants 
is  i  —  ^Q,  and  we  shall  have 

(1)  P,P=v(t-t,)     (Art.  45) 

Let  us  denote  the  coordinates  of  Pq  and  P,  referred  to  any 
convenient  system  of  rectangular  axes,  by  (a:^,  t/q)  and  (x,  y) 
respectively,  so  that 

OM,  =  X,,  M,P,  =  y,.  0M=  X,  MP  =  y, 

'^  /  M^M=  X - x^.  N^N  =  y-y,. 

Since  P  moves  with  a  constant  velocity  v  along  the  line 
A  (7,  the  projections,  itf  and  iV,  of  P  upon  the  x-axis  and  ?/-axis 
will  also  move  with  constant  velocities.  Let  us  adopt  the 
suggestive  notations  v^  and  Vy  for  these  constant  velocities. 
They  are  called  the  a;-component  and  ^/-component  of  the 
velocity  of  P  and,  according  to  Art.  26,  we  have  the  relation 

(3)  ^;2  =  ^^2+^^2 

between  the  three  velocities  v,  v^.,  and  Vy. 


Akt.  80]  MOTIOX   OX   A   STRAIGHT   LINE  121 

The  same  argument  which  led  us  to  equation  (1)  now 
shows  us  that 

whence  we  obtain,  by  substituting  for  MqM  and  iV^JV  their 
values  from  (2), 

(4)  x  =  xq  +  v^(t  -  to),  :y  =  «/o  +  Vyit  - 1^}. 

If  we  begin  to  count  time  from  the  instant  when  P  is  at 
Pq  (any  convenient  place  in  the  path  of  P),  we  shall  have 
t^  =  0,  and  equations  (4)  may  be  written  more  simply  as 
follows 

(5)  x  =  XQ-\-v,t   ij  =  y^  +  vj,. 

Equations  (4)  or  (5)  enable  us  to  calculate  the  coordinates 
(r,  ?/)  of  a  point  P,  which  is  moving  with  a  constant  velocity 
alon;/  a  straight  line. 

These  equations  are  of  great  importance  in  mechanics. 
According  to  one  of  Newton's  laws  of  motion,  they  repre- 
sent the  motion  of  a  body  upon  which  no  forces  are  acting. 
Such  a  body  will  not  move  at  all  if  it  was  at  rest  to  begin 
with,  in  which  case  v^  and  Vy  are  both  equal  to  zero.  But  if 
in  some  way,  before  our  study  of  the  motion  began,  the  body 
had  received  an  impulse  giving  it  a  certain  velocity,  New- 
ton's law  asserts  that  it  will  continue  to  move  with  this  same 
velocity  along  a  straight  line,  just  as  long  as  there  is  no  force 
tending  to  increase  or  diminish  its  velocity  or  to  pull  the 
body  out  of  its  rectilinear  path. 

Ordinary  experience  at  first  sight  seems  to  contradict  this  law.  This 
is  due  to  the  fact  that  we  never  actually  see  a  body  upon  which  no  forces 
are  acting.  If  such  omnipresent  forces  as  gravity,  friction,  resistance  of 
the  air,  etc.,  are  at  least  partially  removed  or  balanced  in  skillfully  ar- 
ranged experiments,  the  contradiction  with  experience  is  found  to  dis- 
appear. 

80.*  Force.  Tf  a  stone  is  thrown  into  the  air,  it  does  not 
describe  a  rectilinear  i)ath  with  constant  velocity.  Accord- 
ing to  Newton's  law  just  quoted  there  must  therefore  be  a 

*  Article  80  may  be  omitted  without  interrupting  the  continuity. 


122  QUADRATIC   FUNCTIONS  [Art.  80 

force  acting  upon  it.  This  force  is  known  as  gravity  and  is 
always  directed  downward.  If  a  stone  be  attached  to  a 
coiled  spring,  gravity  will  cause  the  stone  to  stretch  the 
spring.  Now  we  may  also  stretch  the  spring  by  our  own 
muscular  efforts,  that  is,  by  means  of  our  muscular  force, 
and  this  is  the  reason  that  we  speak  of  gravity  as  being  a 
force.  It  produces  the  same  kind  of  effect  upon  the  stone  as 
though  we  were  pulling  it  down  with  a  certain  muscular  force. 
We  may  use  such  a  coiled  spring  to  compare  and  measure 
forces.  After  we  have  chosen  a  certain  coiled  spring  as  stand- 
ard, we  may  agree  that  the  unit  of  force  is  that  which 
stretches  the  spring  by  one  millimeter.  We  may  then  say 
that  the  numerical  measure  of  a  certain  force  is  F^  if  that 
force  stretches  the  spring  F  millimeters.*  By  measuring 
the  stretching  effect  of  various  masses  upon  the  spring,  the 
following  experimental  law  is  found.  The  force  which  grav- 
ity exerts  upon  a  body  is  proportional  to  the  mass  of  the  body. 
But  the  force  which  gravity  exerts  upon  a  body  is  called  its 
iveight.  Therefore,  the  iveight  of  a  body  is  proportional  to  its 
mass. 

Thus,  if  one  body  has  twice  the  mass  of  another,  it  will  also  have 
twice  the  weight.  But  the  weight  of  a  body  is  not  the  same  thing  as  its 
mass.  The  mass  is  a  number  which  tells  us  how  many  bodies  of  unit 
mass  it  takes  to  counterbalance  a  given  body  on  a  scale  with  equal  arms. 
(See  Art.  43.)  The  weight  is  the  numerical  measure  of  the  force  which 
gravity  exerts  upon  the  body,  and  may  be  measured  by  the  stretching 
effect  of  the  body  upon  a  coiled  spring.  The  mass  of  a  body  remains  the 
same  when  the  body  is  carried  from  one  place  to  any  other  in  the  uni- 
verse. Its  weight,  that  is,  its  stretching  effect,  would  change  if  it  were 
carried  far  away  from  the  earth,  since  the  attractive  force  of  the  earth 
(which  is  the  cause  of  gravity)  becomes  less  as  the  distance  from  the 
earth's  center  increases. 

Let  us  now  attach  a  unit  of  mass  (a  gram  perhaps)  to  our 
standard  coiled  spring.  Let  the  number  of  millimeters  by 
which  this  unit  of  mass  stretches  the   spring  be  called  /. 

*  The  unit  of  force  introduced  in  this  way  is  altogether  arbitrary.  It  is  used 
here  temporarily,  merely  for  the  purposes  of  exi)lanation,  aiid  will  be  replaced 
by  the  customary  units  of  force  presently. 


Art.  80]  FORCE  123 

Then/  represents  the  weight  of  a  unit  of  mass  measured  in 
terms  of  the  temporary  unit  of  force  chosen  above.  (Our 
temporary  unit  of  force  is  the  force  which  stretches  the 
standard  spring  by  one  millimeter.)  Since  the  weight  of  a 
body  is  proportional  to  its  mass,  the  weight  of  a  body  of 
mass  m  will  then  be  mf.  In  other  words :  mf  will  he  the 
measure  of  the  gravitational  force  which  pulls  a  body  of  mass  m 
doivnward,  if  f  is  the  measure  of  the  corresponding  force  for  a 
body  whose  mass  is  equal  to  unity. 

A  second  one  of  Newton's  laws  of  motion  says  that  the 
effect  of  a  force  upon  a  moving  body  is  an  acceleration  ;  that 
is,  a  change  either  in  tlie  magnitude  or  direction  of  its  veloc- 
ity, or  both.  More  specifically  this  law  may  be  stated  as 
follows  : 

1.  Tlie  acceleration  produced  by  a  force  is  always  in  the 
direction  of  the  force. 

2.  The  mag)iitude  of  the  acceleration  is  directly  proportional 
to  the  magnitude  of  the  force. 

3.  The  magnitude  of  the  acceleration  is  inversely  propor- 
tional to  the  mass  of  the  body  upon  which  it  is  acting. 

Thus,  the  acceleration  due  to  gravity  is  always  directed  vertically 
downward,  so  that  the  horizontal  component  of  the  motion  is  not  altered 
at  all  by  the  effect  of  gravity.  If  one  of  two  forces  is  twice  as  great  as 
the  other,  it  will  produce  twice  as  great  an  acceleration,  if  the  body  upon 
which  the  forces  are  acting  remains  the  same.  Finally,  the  same  force, 
acting  upon  two  different  masses,  one  twice  as  great  as  the  other,  -will 
produce  an  acceleration  ujjon  the  greater  mass  just  half  as  great  as  that 
which  it  produces  upon  the  smaller  mass. 

If  we  use  the  language  of  variation,  we  may  state  this 
law  as  follows.  The  direction  of  the  acceleration  produced 
by  a  force  upon  a  body  is  the  same  as  that  of  the  force.  If 
^represents  the  magnitude  of  the  force,  m  that  of  the  mass 
upon  which  it  is  acting,  and  a  that  of  the  acceleration  which 
it  produces,  then  a  varies  directly  as  F  and  inversely  as  m. 
Thus  we  have 

(1)  a  =  k-, 

m 


124  QUADRATIC   FUNCTIONS  [Art.  80 

where  k  is  the  constant  of  variation.  Since  a  reduces  to  k 
when  F  and  m  are  each  equated  to  unity,  it  follows  that  k  is 
the  acceleration  produced  by  a  unit  force  acting  upon  a  unit 
mass.  Consequently  the  numerical  value  of  k  depends  upon 
our  choice  of  the  units  of  force  and  mass,  and  may  be  altered 
by  changing  to  different  units.  We  now  propose  to  do  this 
in  such  a  way  as  to  make  k  equal  to  unity,  so  that  (1) 
becomes 

F 

(2)  a  =  —OYF=ma. 

m 

To  insure  the  validity  of  these  equations  (2),  ive  must  define  as 
unit  of  force  that  force  tvhich,  acting  upon  a  unit  of  mass,  pro- 
duces a  unit  of  acceleration.  For  equations  (2)  give  F=l 
when  m  =  l  and  a  =  1. 

Since  w  is  a  mass  and  a  an  acceleration  whose  dimension 
is  L/T^  (Art.  46),  equation  (2)  shows  that  the  dimension  of 
a  force  is  given  by  the  symbol  ML/  T^. 

We  proceed  to  apply  equation  (2)  to  the  action  of  gravity. 
The  downward  force  exerted  by  gravity  upon  a  body  of 
mass  m  was  found  to  be  mf  if  /  denoted  the  force  exerted 
by  gravity  upon  a  body  of  mass  unity.  To  be  sure,  we 
based  our  justification  of  this  fact  upon  measurements  in 
which  we  used  our  first  unit  of  force  (the  force  which 
stretches  the  standard  spring  by  one  millimeter).  But  this 
relation  will  still  remain  true  whatever  unit  of  force  we  may 
be  using,  the  only  effect  of  a  change  in  the  unit  of  force 
being  to  multiply/ and  mfhj  the  same  factor.  According 
to  (2)  the  acceleration  produced  by  gravity  upon  a  body  of 
mass  m  will  therefore  be 

fm 


m 


=/. 


But  the  acceleration  produced  by  gravity  is  usually  denoted 
by  g  (Art.  48)  and  may  be  measured  by  simple  experiments, 
as  has  already  been  pointed  out  in  Art.  48.  If  we  use  the 
foot  and  second  as  units  of  length  and  time  we  have  very 
nearly  ^^  =  32.2.     If  we  use  the  unit  of  force  implied  by 


Akt  8U]  force  125 

equations  (2), /'will  therefore  also  be  equal  to  32.2.  Thus 
we  have  the  following  result: 

If  the  foot  and  second  are  taken  as  units  of  length  and  time, 
and  if  the  unit  of  force  is  that  force  which  produces  a  unit 
acceleration  when  acting  upon  a  unit  of  mass,  then  the  force 
with  which  gravity/  pulls  a  unit  of  mass  downward  will  be  equal 
to  82.2  of  these  force  units. 

If  we  denote  the  number  o2.2,  as  is  customary,  by  g, 
equation  (2)  shows  that  the  tveight  tv  of  a  mass  m  is  equal 
to  mg,  or 

(3)  w  =  mg, 

since  the  weight  is  the  force  which  gravity  exerts  upon  the 
mass.  Moreover  this  equation  will  hold  only  if  the  weight 
is  expressed  in  terms  of  a  unit  of  force  which  is  related  to  the 
unit  of  mass  in  the  manner  just  indicated. 

So  far  we  have  not  specified  any  particular  unit  of  mass, 
and  therefore  no  unit  of  force  has  as  yet  been  defined 
specifically.  Let  us  take  the  pound  as  unit  of  mass.  Then 
the  unit  of  force  will  be  that  force  which,  acting  upon  a  mass 
of  one  pound,  will  produce  an  acceleration  of  one  foot  per 
second  per  second.  This  unit  of  force  is  called  a  poundal. 
Clearly  the  pound  as  a  unit  of  mass,  and  the  poundal  as  a 
unit  of  force  are  very  different  things.  Finally,  the  weight 
of  a  pound  is  different  from  either  of  these.  The  weight  of 
a  pound  according  to  (3)  is  equal  to  32.2  of  those  force  units 
which  we  have  agreed  to  call  poundals.  Since  tliis  weight 
is  a  force,  it  may  also  be  used  as  a  unit  of  force.  This  unit 
of  force,  the  weight  of  a  pound,  is  called  a  pound  of  force. 
Thus 

(4)  one  pound  of  force  =  32.2  poundals 

where  one  poundal  is  the  force  which,  acting  upon  a  mass  of  one 
pound,  produces  an  acceleration  of  one  foot  per  second  per 
second. 

By  means  of  (4)  it  is  easy  to  convert  forces  expressed  in 
pounds  into  poundals  and  vice  versa. 


126 


QUADRATIC   FUNCTIONS 


[Art.  81 


Fig.  39 


81.   Motion  of  a  projectile  under  the  influence  of  gravity. 

In   Fig.    39    let   the    positive   ^-axis  be  directed  vertically 
upward,  and  let  the  a;-axis  be  a  horizontal  line  in  the  plane 

of  the  curved  path 
described  by  a  pro- 
jectile which  starts 
from  the  point  Pq 
whose  coordinates 
are  (%  ?/y).  Let 
V^  and  Vy  be  the 
components  of  the 
initial  velocity  V 
and  let  us  count  time  (in  seconds)  from  the  moment  in  which 
the  projectile  begins  its  flight.  If  gravity  were  not  acting, 
the  projectile  would  after  t  seconds  reach  a  point  whose  co- 
ordinates are  (compare  equations  (5)  Art.  79) 

The  action  of  gravity  has  no  effect  upon  the  first  of 
these  two  equations,  which  represents  the  horizontal  com- 
ponent of  the  motion ;  but  it  will  cause  the  projectile  to 
be  in  a  lower  position  at  every  instant  than  it  would  oc- 
cupy if  gravity  were  not  acting.  This  eifect  of  gravity 
is  taken  care  of  by  addition  of  the  term  —  \  gt"^  to  the 
right  member  of  the  second  equation.*  Thus  we  obtain  the 
equations 


(1) 


x  =  Xq+  Vj,  y  =  yQ+Vyt- i yt\ 


which  enable  us  to  compute  the  coordinates  of  the  projectile,  t 
seconds  after  the  beginning  of  its  flight,  provided  that  it  has  not 
struck  an  obstacle  in  the  meantime. 

In  using  tliese  equations,  V^  and  Vy  are  positive  or  negative  according 
as  these  velocity  components  have  the  directions  of  the  positive  or  nega- 
tive X-  and  7/-axes. 

*  We  are  not  attempting  to  prove  this  statement.  The  term  —  ^  gt^  represents 
the  distance  through  which  the  projectile  would  have  fallen  in  the  time  t  if  it  had 
been  dropped  i'roiu  Fq  without  any  initial  velocity. 


Art.  81]  MOTION   OF   A   PROJECTILE  127 

The  velocity  components,  v^.  and  I'y,  of  the  projectile  at  the 
time  t  are  given  by 

(2)  t^.=  r„  v,  =  v,-ot. 

The  truth  of  these  equations  follows  immediately  from 
Newton's  laws  of  motion  if  we.  regard  the  strength  of 
gravity  as  being  the  same  at  every  point  of  the  path  of  the 
projectile. 

EXERCISE  XXXI 

1.  A  ball  is  thrown  vertically  upward  with  a  velocity  of  50  feet  per 
second.  How  high  will  it  be  above  its  starting  point  at  the  end  of  2 
seconds  ? 

Hint.  We  may  choose  the  origin  at  the  point  P^^  so  that  x^  =  ij^  =  0. 
In  this  case,  moreover,  V^  =  0,  Vy  =  +  50  feet  per  second. 

2.  In  how  many  seconds  will  the  ball  of  Ex.  1  reach  its  greatest 
height,  and  how  high  will  it  be  at  that  time? 

Hint.     Find  the  maxiniuni  of  y. 

3.  At  what  time  will  the  ball  of  Ex.  1  be  30  feet  from  the  ground? 
Why  do  you  obtain  two  answers? 

4.  How  long  will  it  take  the  ball  of  Ex.  1  to  return  to  its  initial 
position  ? 

Hint.     Equate  y  to  zero. 

5.  The  initial  velocity  of  the  projectile  from  an  8.8  centimeter  Krupp 
gun  is  442  meters  per  second.  If  the  barrel  of  the  gun  makes  an  angle 
of  45°  with  the  horizontal  plane  upon  which  it  stands,  at  what  distance 
from  the  gun  will  the  projectile  strike  this  same  horizontal  plane  ? 

6.  Show  that  the  path  described  by  a  projectile  according  to  equations 
(1)  of  Art.  81,  is  a  parabola. 

Hint.  Eliminate  t  between  the  two  equations  (1),  and  compare  the 
resulting  equation  between  x  and  y  to  the  equations  considered  in 
Art.  65. 

7.  Find  the  highest  point  of  the  path  of  the  projectile  described  in 
Ex.  5. 

8.  Find  formulas  for  the  coordinates  of  the  highest  point  of  the  path 
of  a  projectile,  using  the  notations  of  equations  (1),  Art.  81. 


128  QUADRATIC   FUNCTIONS  [Art.  81 

9.  Using  the  general  equations  (1)  of  Art.  81,  find  a  formula  for 
the  distance  from  the  gun  at  which  a  projectile  will  again  strike  the 
horizontal  plane  of  the  point  P^  from  which  it  begins  its  flight.  (This 
is  the  range.) 

10.  Making  use  of  the  result  of  Ex.  9  show  that  the  range,  for  a  gun 
of  given  power,  is  a  maximum  when  V^  =  V^,  that  is,  when  the  gun  is 
elevated  at  an  angle  of  45°. 

11.  A  force  gives  a  mass  of  10  pounds  an  acceleration  of  3  feet  per 
second  per  second.  Express  the  magnitude  of  this  force  in  pounds  of 
force  and  in  poundals. 

12.  What  acceleration  would  the  force  described  in  Ex.  11  give  to  a 
mass  of  3  pounds  ? 


CHAPTER   IV 

INTEGRAL   RATIONAL    FUNCTIONS   OF   THE   nth   ORDER. 
NUMERICAL    CALCULATION    OF   THEIR  REAL    ZEROS 

82.  Calculation  of  the  numerical  values  of  an  integral  rational 
function.  A  function  is  said  to  he  an  integral  rational  function 
of  the  nth  order,  if  it  can  he  expressed  in  the  form 

(1)  y  =  Ax^  +  ^.r"-i  +  (7x"-2  +  ...  +  Lx  +  M 

where  n  is  a  positive  integer  and  where  the  coefficients  A,  B,  (7, 
•  ••  i,  M  are  constants. 

Thus,  an  integral  rational  function  of  x  is  composed  of  a 
finite  number  of  terms,  each  of  these  terms  being  a  product  of 
a  constant  by  a  power  of  x  with  a  finite  positive  integer  as 
exponent.  The  order  or  degree  of  such  a  function  is  de- 
termined by  the  exponent  of  the  highest  power  of  x  which 
actually  occurs  in  it.  The  linear  and  quadratic  functions, 
discussed  in  the  preceding  chapters,  are  integral  rational 
functions  of  the  first  and  second  order  respectively. 

The  method  used  in  Chapters  II  and  III,  for  representing 
a  linear  or  quadratic  function  geometrically  by  a  graph,  may 
be  applied  to  an  integral  rational  function  of  the  wth  order 
as  well.  It  is  evident  that  the  abscissas  of  those  points 
where  the  graph  crosses  the  2;-axis  will  be  the  real  roots  of 
the  equation  of  the  nth  order  obtained  by  equating  to  zero 
the  function  (1).  Consequently  the  construction  of  the 
gi'aph  of  such  a  function  will  give,  by  inspection,  either  the 
exact  or  the  approximate  values  of  the  real  roots  of  an 
equation  of  the  wth  order. 

Tlie  construction  of  the  graph  of  such  a  function  is  really 
no  more  difficult  than  for  a  linear  or  quadratic  function,  but 

129 


130  INTEGRAL   RATIONAL   FUNCTIONS  [Art.  82 

the  calculations  required  are  a  little  longer.  We  shall  there- 
fore explain  first  a  simple  method  for  calculating  the  value 
of  a  function  of  the  form  (1)  for  a  given  value  of  x. 

It  will  suffice  to  explain  this  method  for  the  case  of  a 
function  of  the  third  order  (a  cubic  function), 

(2)  y  =  A3^-\-  Bx^  -\-  Cx+D. 

To  compute  the  value  of  (2)  for  x=  a^  we  first  multiply  A 
by  a  and  add  B  to  the  product.  Let  the  result  be  called  E, 
so  that 

(3)  Aa+  B  =  E. 

We  then  multiply  E  by  a  and  add  C.  Let  the  result  be 
called  F,  so  that 

(4)  Ea  +  C  =  {Aa  +  By  +  0=Aa^-  +  Ba+  C  =  F. 

We  repeat  this  operation  once  more,  obtaining 

(5)  Fa  +  D=iAa^  +  Ba+C~)a+D  =  Aa^  +  Ba^+Ca  +  D=a. 

But  this  is  precisely  the  quantity  whose  value  we  wished  to 
calculate,  that  is,  Gr  is  the  value  of  the  function  (2)  for 
X  =  a. 

The  calculation  may  therefore  be  arranged  as  follows : 

(6)  A         B         O        D     [a 

Aa       Ea       Fa 

^      ^      a' 

Thus,  if  we  wish  to  calculate  the  value  of  3  a;^  —  4  x^  +  7  x  —  2  for 
x  =  -2,wewrit^       3     -    4     +    7     -    2     [-2 
_    6     +20     -  54 
^^IlO     +27     ^56 

Giving  —  56  as  the  result,  which  may  be  verified  directly. 

It  should  be  remembered,  in  applying  this  method,  that  if 
any  one  of  the  terms  of  Ax^  +  Bx^  -\-  Cx  ■\-  D  i^  absent,  the 
corresponding  column  in  the  calculation  (6)   must   not   be 


Art.  83]  THE   FUNCTIONAL   NOTATION  131 

omitted.  The  missing  term  should  be  written  with  zero  as 
its  coefficient. 

This  proce^',  which  is  applicable  to  integral  rational 
functions  of  any  degree,  is  very  important  and  should  be 
used  in  connection  with  the  following  examples. 

EXERCISE  XXXII 

Make  graphs  for  the  following  functions,  and  find  either  exact  or 
approximate  values  of  their  zeros  by  inspecting  the  graph.  (Approxi- 
mations to  the  nearest  half  unit  are  sufficient.) 

1.  y  =  x^.  4.    ^  =  —  x^.  1.    y  =  x^  —  1. 

2.  y  =  2x^.  5.    y  =  -2x^.  8.    ^  =  r^  +  1. 

3.  3/  =  3  x3.  G.  y  =  -?,  x".  9.  ^  =  x3  -  8. 

10.  y  =  x3  -  6  x2  +  8  X. 

11.  y  =  x3  -  3  x2  -  X  +  3. 

12.  y  =  2x3  -  .5x2-  2x +5. 

83.  The  functional  notation.  It  is  often  convenient  to 
have  a  short  notation  for  a  given  function  of  x.  Thus  we 
may  write  ^  _    „       .    „ 

and  we  can  then  express  the  result  of  the  calculation  at  the 
end  of  Art.  82  very  briefly  by  saying  that 

/(-2)=-56. 

The  symbol  /(a;)  is  read  the  f -function  of  x.  In  any  specific 
example /(a;)  stands  for  a  definitely  given  function  of  x.  In 
different  examples  /(a:)  may  stand  for  different  functions. 
If  several  different  functions  occur  in  the  same  example,  we 
may  denote  them  by /(a;),  g(x).,  A  (a:),  etc.,  or  by  /i(a;), 
/2(x-),  fz(x)-,  etc.  The  symbol  f(x)  does  not  mean  /  multi- 
plied by  X. 

EXERCISE  XXXIII 

1.  Given/(x)=  x-^  -  x^ -i-  x  -  1.  Find/(())./(l),/(2),/(3),/(-l), 
/(-2),/(-3). 

2.  Given  /(x)  =  2 x^  -  5 x-^  -  2 x  -h  5.  Find  /(O),  /(I), /(2), /(;$), 
/(-!),/(- 2), /(-:}). 


132  INTEGRAL   RATIONAL   FUNCTIONS  [Art.  84 

3.  Given/(x)=3x  +  5.    Find /(x^),  [/(x)p,/[/(x)], /Q, /(x^). 

4.  Given   y  =  f{z)=  z^  +  ?>,  z  =g  {x)=  ^  x^  -  x  +  5..     Find  f\jg  (x)]. 

5.  Given    /(x)  =  x^  +  5,    ^(x)=  3  x'^  -  x  +  5.       Find  /[^^(x)]     and 

6.  Show  that  a  linear  function  of  a  linear  function  of  x  is  again  a 
linear  function  of  x. 

7.  Show  that  a  linear  function  of  a  quadratic  function  of  x  is  a  quad- 
ratic function ;  and  that  a  quadratic  function  of  a  linear  function  of  x 
is  a  quadratic  function  of  x. 

8.  Show  that  a  quadratic  function  of  a  quadratic  function  of  x  is  a 
function  of  the  fourth  degree. 

84.  The  factor  theorem.  We  have  already  noticed  that  a 
quadratic  function  which  has  a:  —  a  as  a  factor  vanishes  for 
2J  =  a,  and  conversely,  that  if  a  quadratic  function  vanishes 
for  x=  a^  then  it  has  x  —  a  as  a  factor.  (See  Art.  68.) 
We  shall  now  show  that  a  corresponding  theorem,  known 
as  the  factor  theorem,  holds  for  integral  rational  functions 
of  any  order. 

If  X  —  a  is  a  factor  of  an  integral  rational  function  f  (x^,  the 
function  will  assume  the  valUe  zero  when  x  is  equated  to  a.  Con- 
versely^ if  such  a  function  f  (^x)  becomes  equal  to  zero  for  x  =■  a^ 
thenf(x)  has  x  —  a  as  a  factor. 

The  proof  of  the  first  part  of  this  theorem  is  immediate. 
If /(a;)  has  a;  —  a  as  a  factor,  we  may  write 

f(x)  =  {x-a>^g(x), 

where  ^(2;),  the  other  factor  of /(a;),  is  an  integral  rational 
function  of  x  whose  degree  will  be  less  by  a  unit  than  that 
oif(x).  If  in  this  equation,  we  put  a:  =  a,  the  factor  x  —  a 
becomes  equal  to  zero,  and  the  other  factor  becomes  equal 
to^(«),  which  will  be  some  finite  number.  Consequently 
the  product  will  vanish,  so  that  /(a)  =  0,  as  we  wished  to 
prove. 

To  prove  the  converse,  let 

n )  fix)  =  Ax^  +  5a:"-i  +  6V-2  +  ...  ^Lx^^Mx^N. 


Art.  84]  THE   FACTOR   THEOREM  133 

According  to  the  hypothesis,  the  value  of  this  function  for 
x=  a^  that  is,/(rt),  is  equal  to  zero.     Therefore  we  have 

(2)  /(a)=^a"  +  ^a"-i+  Ca"-2  +  ...  +La^  +  Ma  +  N=Q. 
We  may  therefore  write 

(3)  /(2-)=/(2^)-/(a)=^(2;»-a»)  +  5(x»-i-a'»-i)4-  ••. 

+  Lix^  -  0^)+  M(x-  a). 

Each  of  the  last  two  binomial  terms  of  (3)  obviously  has 
a:  —  a  as  a  factor:  We  shall  show  immediately  that  the  same 
thing  is  true  of  each  of  the  other  binomial  terms  of  (3). 
Consequently,  f(x^  has  a:  —  a  as  a  factor,  as  was  to  be  proved. 

In  order  to  complete  the  proof  of  this  theorem  it  only  re- 
mains to  show  that,  for  every  value  of  the  positive  integer  n, 
z"  —  a"  has  x  —  a  as  a  factor. 

We  know  that  this  is  true  for  n  =  1  and  for  n  =  2,  since 

a:  —  a  =  (a;  —  a)  •  1 ,  x^  —  a^  =  (x  —  a)(x  +  a^. 

It  may  be  verified  easily  that  it  is  also  true  for  7i  =  3,  since 

x^  —  a^  =  (^x  —  a)  {x^  +  ax  +  a^), 

as  may  be  seen  by  performing  the  multiplication  on  the  right 
member  and  simplifying.  To  prove  that  this  theorem  is  true 
for  all  values  of  w,  we  first  prove  the  following  lemma,  or 
auxiliary  theorem. 

If  x^  —  a^  ha^  x  —  a  as  a  factor^  so  does  x^""^  —  a'^'^^. 
Proof.      \Vc  may  write 

^k¥\  _   ^A+l  _  j.k  +  1  _   ^kj.    _|_    ,^kJ.  _  ^A  +  l  __  ^^J^k  _   (ikyy  _j_  (ik^j.  _   ^-^^ 

Since  the  last  term,  rt*(a-  — a),  has  x  —  a  as  a  factor,  the  whole 
right  member,  which  is  equal  to  x^  ^  —  a''"'\  will  have  x—a 
as  a  factor  if  x'^  —  a'''  has  such  a  factor,  lint  this  remark 
proves  tlie  lemma. 

Now  2;2  —  a^  has  a- —  a  as  a  factor.  A  first  application  of 
the  lemma  (for  ^  =  2)  sliows  that  ;r^+^  —  a-'*"^  or  .x^  —  a^  also 


134  INTEGRAL   RATIONAL   FUNCTIONS  [Art.  85 

has  X—  a  as  a  factor.  A  second  application  of  the  lemma 
(to  the  case  A;  =  3)  shows  that  a;*  —  a*  has  a;  —  a  as  a  factor. 
We  may  proceed  in  this  way  until  we  reach  re"  —  a",  thus 
proving  that  x^  —  a"  has  a;  —  a  as  a  factor. 

The  method  of  proof  just  employed  is  called  the  method 
of  mathematical  induction,  and  is  very  important  in  all  parts 
of  mathematics.  We  shall  have  occasion  to  apply  this 
method  frequently  during  this  course. 

85.  The  remainder  theorem.  The  following  theorem, 
known  as  the  remainder  theorem,  includes  tXvd  factor  theorem 
as  a  special  case. 

If  an  integral  rational  function  f(^x)  be  divided  hy  x  —  a 
until  a  remainder  independent  of  x  is  obtained^  this  remainder 
is  equal  tof(^a^,  the  value  of  the  function  f(^x^  for  x  =  a. 

Proof.  Carry  out  the  process  of  dividing  /(a^)  hjx  —  a 
until  we  reach  a  remainder  M  independent  of  x,  and  let  the 
quotient  obtained  by  this  division  be  called  Qi^x).  Accord- 
ing to  the  definition  of  the  terms  division,  quotient,  and 
remainder  (see  Art.  4),  this  means  that  we  shall  have 

(1)  flx)=Q(:xX^-a)+B, 

where  i^  is  a  constant  whose  value  does  not  depend  upon  x,  and 
where  Q^x}  will  be  of  degree  w  —  1  if  /(a^)  is  of  degree  n.* 

Let  us  now  put  ;r  =  a  in  (1).  Since  Q{x^  is  an  integral 
rational  function  of  ;r,  (?(«)  is  a  finite  number,  and  we  find 
from  (1) 

(2)  /(a)=  Q(a)  .0  +  B=E. 

But  (2)  is  nothing  more  or  less  than  the  remainder  theorem 
which  we  wished  to  prove. 

In  the  particular  case  when  M=0,  we  obtain  from  (1) 
and  (2)  the  factor  theorem  ;  saying  that  if  /(a)  =  0,  then 
/(a;)  has  X  —  a  as  a  factor,  and  conversely.  Thus  we  have 
found  incidentally  a  new  proof  of  the  factor  theorem. 

*  This  process  shows  that /(x)  may  be  written  in  the  form  (1)  and  does 
not  involve  any  actual  division  until  we  write 


Art.  SG]  synthetic    DIVISION  135 

86.  Synthetic  division.  Tlie  remainder  theorem  shows  us 
lliut  if  we  divide  f(^x)  by  a;  —  a,  the  remainder  is  equal  to 
/(a).  But  in  Art.  82  we  found  a  convenient  method  for 
calculating  the  value  of /(a).  We  may  therefore  use  this 
same  method  for  calculating  the  remainder  in  the  division 
oif(x)  by  X—  a.  But  this  calculation  will  at  the  same  time 
give  us  the  value  of  the  quotient.  To  see  this,  let  us  apply 
the  ordinary  process  of  long  division  to  the  problem  of  divid- 
ing Ax^  +  Bx^  +  Cx  +  D  by  j- —  a,  and  then  let  us  compare 
with  Art.  82.      We  find 

Ai^  +  Bx'^  +  Ox  +  D  x-a 
Ax^  -  Aax^  I  Ai^  +  Ex  +  F 

Ux^'+  Ox 
Bx^-  -  Eax 

Fx  +  D 
Fx  -  Fa 

Fa  +  B=a  =  R 

The  term  Fx^  is  first « obtained  in- the  form  Aax^ -{- Bx^ 
=  {Aa  +  B}x'^,  but  according  to  the  notation  used  in  (8) 
Art.  82,  this  is  the  same  as  Fx^.  Similarly  the  term  Fx 
arises  from  (Fa  +  0)x  which,  according  to  (4)  Art.  82,  is 
equal  to  Fx. 

If  now  we  write  down  once  more  the  scheme  for  the 
calculation  explained  in  Art.  82,  namely 

A         B         a        B  [a 

Aa       Fa       Fa 

A        F        F        B~=f(a), 

we  notice  that  the  first  three  numbers  of  the  last  line,  A,  E, 
and  F,  are  the  coefficients  of  x\  x,  and  1,  in  the  quotient, 
whereas  M  is  the  remainder. 

As  applied  to  the  example  f(x)  =  3  x^  —  i  x'^  +  7  x  —  2,  which  was 
used  at  the  end  of  Art.  82  as  an  illustration,  we  see  from  the  numbers 
obtained  there  that  the  quotient  obtained,  when  3  x^  —  4  x^  +  7  x  —  2  is 
divided  by  x  —  (-  2)  =  x  +  2,  will  be  S  x^  -  10  x  +  27,  and  the  remainder 
will  be  —  56.  This  calculation  has  been  written  out  in  just  this  form  in 
Art.  82. 


136  INTEGRAL   RATIONAL   FUNCTIONS  [Art.  87 

This  method  of  dividing  an  integral  rational  function  by 
a;  —  a  is  far  more  convenient  than  the  ordinary  method.  It 
is  known  as  the  method  of  synthetic  division.  Synthetic 
division  may  be  performed  according  to  the  following  rule. 

1.  To  divide  f(x)  by  x  —  a,  arrmige  f(x')  in  descending 
powers  of  X. 

2.  Write  the  coefficients  of  /(x)  on  a  horizontal  line,  in  the 
order  ivhich  corresponds  to  the  arrangement  specified  in  No.  1 
of  this  rule.  If  any  power  of  x  is  missing  inf(x).,  supply  that 
power  with  a  zero  coefficient. 

3.  Multiply  the  first  coefficie7it  A  by  a,  write  the  product 
beloiv  the  second  coefficient  B,  and  add.  Multiply  this  sum  E 
by  a,  ivrite  the  product  below  the  third  coefficient  (7,  and  add. 

4.  Proceed  in  this  umy  until  all  of  the  places  in  the  third 
row  except  the  first  are  filled  up,  and  ivrite  down  the  first  coef- 
ficie7it  A  of  f(x')  in  the  first  place  of  the  third  roiv.  The  last 
number  of  the  third  row  will  be  the  retnainder.,  and  the  other 
numbers  in  the  third  row  will  be  the  coefficients  of  the  quotient 
obtained  when  f(x)  is  divided  by  x  —  a. 

EXERCISE  XXXIV 

Find  the  quotient  and  remainder  in  the  following  divisions.  Use  the 
method  of  synthetic  division. 

1.  Divide  2  x^  +  5  a;^  -  7  x  +  1  by  a;  -  3. 

2.  Divide  .3  a.-^  -  7  a;^  +  4  a:  -  5  by  x  +  3. 

3.  Divide  2  a;^  +  x  +  1  by  a'  -  2. 

4.  Divide  x^  —  1  by  x  +  2. 

87.  The  slope  of  the  tangent.  When  we  have  drawn  the 
graph  of  an  integral  rational  function  y  =  f{x)  by  the  method 
of  computing  the  coordinates  of  a  large  number  of  its  points, 
we  can  draw  the  tangent  to  the  curve  at  any  one  of  its  points 
with  some  degree  of  approximation.  We  wish,  however,  to  be 
in  a  position  to  draw  the  tangent  with  greater  accuracy,  and 
this  desire  leads  us  to  adopt  a  precise  detinition  for  a  tangent 
and,  in  this  way,  to  seek  a  precise  method  for  its  determination. 


Art.  87] 


THE   SLOPE   OF   THE   TANGENT 


137 


We  define  a  tangent  to  a  curve  as  follows : 

Let  Pj  be  any  point  on  a  given  curve  {see  Fig.  40),  and  let 
P^  be  a  second  point  (^distinct  from  Pj)  of  the  same  curve.  As 
Pg  approaches  P^  as  a  limit,  the  line  PiP^  (sometimes  called 
a  secant  of  the  curve}  will  turn  around  Pj  as  a  center.  If  the 
secant  approaches  a  limiting  position  P^T  as  P^  approaches 
Pj,  this  limiting  position  of  the  secant  is  called  the  tangent  of 
the  curve  at  P^,  and  J\  is  called  its  point  of  contact. 

This  detiuitiou  ie(iiiires  a  few  words  of  explanation.  In  the  fir.st 
place  it  presiii)poses  the  notion  of  limits  which  the  student  has  discussed 
to  some  extent  in  his  earlier 
courses  in  elementary  algebra 
and  geometry.  A  more  detailed 
discussion  of  this  important 
subject  will  be  given  later  in 
this  book.  (See  Chapter  XV.)* 
In  the  second  place  it  is  essen- 
tial to  remember  that  the  line 
PiP.,  is  regarded  as  unbounded. 
We  are  not  talking  about  the 
line-segnient  Px^r  ^^^li^'l'  <j£ 
course  approaches  zero  when 
Pi  approaches  Pj,  but  of  the 
whole  line  upon  which  this  seg- 
ment is  located.  Finally,  when 
we  say  that  Pg  shall  approach  Pj,  it  should  be  understood  that,  during 
this  approach,  P,  must  always  remain  on  the  given  curve  and  that  it 
must  reTuain  distinct  from  Pi  but  that  it  may  approach  Pj  from  the 
right,  or  from  the  left,  through  positions  which  are  partly  to  the  right 
and  partly  to  the  left  of  Pp  either  continuously  or  by  jumps. 

Let  us  now  denote  the  coordinates  of  Pj  and  P^  by  (.r,  ,y) 
and  (x  +  h,  y  +  k)  respectively,  and  let  the  given  curve  be 
the  graph  of  y  =f{x).  Then,  the  ordinate  of  every  point  of 
the  curve  (the  y  of  that  point)  will  be  equal  to  the /-func- 
tion of  the  abscissa  (the  x  of  that  point).     Thus  we  shall 

have 

y  =/(x),  y+k  =f(x  +  A), 


Fig.  40 


*  If  the  instructor  prefers,  part  or  all  of  this  chapter  on  limits  may  be  inserted 
here  before  proceeding  farther. 


138  INTEGRAL   RATIONAL   FUNCTIONS  [Art.  87 

since  y  +  k  \^  the  ordinate  and  x  -{■  h  the  abscissa  of  P^.     The 
slope  of  the  secant  PiP<y,  will  be,  according  to  Art.  53, 

(1)  ^^  =  y  +  ^-  y  ^f(^-^  ^0  -  ./'C-O . 

X  -\-h—  X  h 

Iq  Fig.  40  we  have 

X  =  OMi,   X  +  h  =  OM^,    h  =  M^M,  =  P^Q, 

y  =  M^P„  y  +  k  =  M^P.-,,  k  =  QP.. 

,,    ,  QPo 

so  that  m  =  v,— ?• 

PiQ 

If  now  we  let  P^  approach  Pj  as  a  limit,  h  will  approach 
the  limit  zero,  and  the  slope  of  the  secant  will  in  general 
approach  as  its  limit  the  slope  of  tlie  tangent  to  the  curve  at 
Py     Thus  we  obtain  the  following  result : 

Theorem  I.  Consider  the  curve  which  is  obtained  as  the 
graph  of  the  function  y  =f(x'),  and  let  P^  he  any  particular 
point  on  that  curve  which  has  a  uniquely  determined  tanqeMt, 
not  parallel  to  the  y-axis.  Then  the  slope  of  the  tangent  of  this 
curve  at  the  point  P^  is  equal  to  the  limit  ivhich  the  quotient 

fCx+h^-f(x) 
h 
approaches^  ivhen  h  approaches  zero,  if  x  denotes  the  abscissa  of 
the  point  P^ 

The  condition  that  the  curve  shall  have  a  Tiniquely  determined  tangent 
at  Pj  has  been  mentioned  so  as  to  exclude  such  points  as  A  in  Fig.  41. 
The  condition  that  the  tangent  shall  not  be  parallel  to 
the  y-axis  has  been  put  in,  because  a  line  parallel  to  the 
^\^  y-axis  cannot  be  said  to  have  a  slope.     (See  Art.  53.) 

-■1  In  neither  of  these  cases  has  the  quotient 


"*  /(x+A)-/(x) 

Fig.  41  ^ 

a  unique  finite  limit.     However,  we  shall  see  later  that  these  exceptional 
cases  never  occur  iif(x)  is  an  integral  rational  function  of  x. 

Unless  the  graph  of  y=:f(x)  is  a  straight  line,  that  is, 
unless  /(a;)  happens  to  be  a  linear  function,  the  slope  of  the 
tangent  will  be  different  for  different  points  of  the  curve. 


Art.  ST]  DERIVATIVE   OF    A    FUNCTION  139 

Consequently  tlie  limit  mentioned  in  Theorem  I  will,  in 
general,  be  a  variable  function  of  x.  This  new  function  of  x 
derived  from  the  function  /(a;)  by  a  definite  limit  process,  is 
called  the  derivative  of /(x),  and  is  usually  denoted  by /'(a;). 
We  may  therefore  write  the  following  statement  which  may 
be  regarded  as  a  definition  of  the  derivative  of  a  given  function. 

TJie  function  fX-^)  ivhich  is  obtained  from  a  given  function 
f  (i-)  hy  first  forminy  the  quotient 

fix  +  /Q  -f(x-) 
h 

and  then  ohtaininy  the  limit  tvhich  this  quotient  approaches 
when  h  approaches  zero^  is  called  the  derivative  off(x'). 
In  symbols  this  definition  may  be  formulated  as  follows  : 

(2)  f'(x)=lim-L(^±JO-f('^\ 

where  the  right  member  of  this  equation  is  to  be  read :  limit  of 
the  fraction,  f{j-  +  h)—f(x)  divided  by  h,  as  h  approaches 
zero. 

If  we  use  this  terminology,  Theorem  I  may  now  be  re- 
formulated as  follows  : 

Theorem  II.  Let  us  consider  the  graph  of  the  function 
y  =f{x)  and  letf'(x)  be  the  derivative  of  f{x).  If  we  give  a 
definite  numerical  value  to  x  and  compute  the  value  of  the 
derivative  f  {x)  for  this  value  of  x,  the  result  ivill  he  equal  to  the 
slope  of  that  tangent  of  the  curve  whose  point  of  contact  has  this 
given  value  of  x  as  its  abscissa,  provided  that  the  point  of  eon- 
tact  is  one  where  the  curve  actually  possesses  a  unique  tangent 
which  is  not  parallel  to  the  y-axis. 

The  notion  of  a  derivative  was  introduced  into  mathematics,  nearly 
simultaneously  and  probably  independently,  by  the  great  English  mathe- 
matician and  physicist  Siu  Isaac  Newton  (1643-1727)  and  by  the 
famous  German  philosopher  and  mathematician  G.  VV.  Leibniz  (1646- 
171G).  This  notion  is  the  fundamental  concept  of  the  Differential  Cal- 
culus. 


140 


INTEGRAL   RATIONAL   FUNCTIONS 


[Art.  87 


The    following    illustration    will    make    this    process    clearer.     Let 
f{x)  —  x^,  so  that  the  graph  considered  will  be  tiie  parabola  obtained 
by  making  the  graph  of 

y  =  A 

and  let  us  compute  the  slope  of  that  tangent  of 
the  parabola  whose  point  of  contact  has  the  ab- 
scissa X  =  3/2  and  whose  ordinate  will  therefore 
hey=  (3/2)^  =  9/4. 
AVe  have,  in  this  case, 


/(f)  =  I, 


+  2  (I)  A  +  /<2, 


so  that  according  to  (1) 


=  ^  +  h 


is  the  slope  of  tliat  secant  determined  by  those  two  points  on  the  parabola 
whose  abscissas  are  equal  to  |  and  |  +  h  respectively.  As  h  approaches 
zero,  m  will  approach  the  limit  3.  Therefore  the  slope  of  the  tangent  of 
the  parabola  at  the  point  (3/2,  9/4)  is  equal  to  3. 

More  generally,  we  may,  according  to  Theorem  II,  by  finding  the 
derivative  of  /'(x)  =  a;-,  compute  the  slope  of  the  tangent  at  cnui  point  of 
the  parabola.     This  slope  will  be  equal  to 

f'(j-)  =  lim*^— ^^ J- —    ^      =  lim  -^^ ^ , 


hm — Z J: =  ]i,n -T — =  ii,„  (^x  +  h), 


f'(x) 

or  finally  y(x)=2x. 

Thus,  if  we  consider  any  point  on  the  paral)nla  whose  abscissa  has  a 
given  value  x,  and  construct  the  tangent  at  that  point,  the  slope  of  this 
tangent  will  be  equal  to  2  x. 

In  the  particular  case  when  x  =  3/2  we  find  the  slope  to  be  twice  3/2 
or  3  as  before. 

EXERCISE  XXXV 

By  making  use  of  the  definition  of  the  derivative  as  given  in  Art.  87, 
find  the  derivatives  of  the  following  functions : 

1.    1/  =  3  X  +  5.  3.    ?/  =  f)  x-  +  3  X  +  5. 


2.   ^  =  5  x^. 


AuT.  88]  THE   BINOMIAL   THEOREM  141 

5.  //  =  7  x-''.  7.   y  =  j3  _|_  5  2;2_ 

6.  //  =  X*.  8.    ij  =  x^  —  5  x^. 

9.  Compute  the  slope  of  the  straight  line  which  touches  the  graph 
of  7j  =  3  x'^  —  2  X  +  4  at  that  point  of  the  curve  whose  abscissa  is 
equal  to  2. 

10.  Compute  the  slope  of  tlie  straight  line  which  touches  the  graph 
of  //  =  |x*^  —  4  at  that  jwint  of  the  curve  whose  abscissa  is  equal  to  —  2. 

88.  The  binomial  theorem.  W^e  ure,  in  this  chapter,  stud;y'- 
ing  integral  rational  functions,  that  is,  functions  of  the  form 

y  =fix)  =  Ax"  +  Bx'"'  +  ...■+Lx+  M, 

and  the  graphs  of  such  functions.  If  we  wish  to  be  in  a 
position  to  draw  the  tangent  to  snch  a  graph  at  any  one  of 
its  points  we  must,  according  to  Art.  87,  calculate  the 
derivative  of /(a;).  In  order  to  do  this,  we  must  first  com- 
pute the  value  of 

f(x  +  h)  =  A(x  +  hy  +  B(x  +  hY-^  +  •••  +  L(x  +  h)  +  M, 

then  form  the  difference /(a;  +  h^  —/(a:),  divide  this  differ- 
ence by  A,  and  finally  evaluate  tlie  limit  which  this  quotient 
approaches  when  h  approaches  zero.  But  before  we  can 
carry  out  this  program,  we  must  first  learn  how  to  calculate 
the  terms  {x  +  A)",  (x  -f  hy~\  etc.,  which  occur  in  the  above 
expression  for /(a;  -f  A). 

The  general  formula  for  (^x  -f  A)"  which  wc  shall  now 
derive  is  commonly  known  as  the  binomial  theorem  and  is 
due  to  Sir  Isaac  Newton.  We  shall  prove  it  by  making 
use  of  tlie  method  of  mathematical  induction  which  we  have 
already  used  in  Art.  84. 

It  is  easy  to  verify,  by  actual  multiplication,  that 
(x  +  hy  =  X  +  h, 
(x  +  h)'^-  X-  +  2  xh  +  h% 
(1)  (x  +  hy-i  =  x8  +  3  x2A  +  3  xh^  +  h\ 

(x+hy  =  x*  +  4x'^h  +  Ci  x'^h-  +  4  xP  +  /(•», 

(x  +  hy  --^  x^  +  5  x*h  +  10  x^-  +  10  xW^  +  .1  xh*  +  h^. 


142  INTEGRAL   RATIONAL   FUNCTIONS  [Art.  88 

Careful  inspection  of  tliese  special  cases  suggests  that  the  following 
laws  are  probably  true. 

1.  The  expansion  of  (.r  +  //)"  contains  n  +  1  terms. 

2.  The  first  term  will  be  x". 

3.  The  second  term  will  be  ?u;"~Vt. 

4.  The  tliird  term  will  be  "-^^^—^ — ^x"~-h-.     The  numerical  coefficient 

of  this  term  may  be  obtained  from  the  second  term  by  multiplying  its 
coefficient  /(  l)y  the  exponent  of  x  in  that  term,  namely  n  —  1,  and  divid- 
ing the  product  by  2,  which  is  the  number  of  the  term. 

5.  The  fourth  term  will  be 

"(^-l)("--)x"-3R 
1  •2-3 

The  numerical  coefficient  of  the  fourth  term  may  be  obtained  from  the 
third  term  by  multiplying  its  coefficient,  "^"  ~^  ^ ,  by  the  exponent  of 

.X-  in  that  teriii,  namely  7i  -  2,  and  dividing  by  3,  the  number  of  the 
term. 

6.  If  we  multiply  the  coefficient  of  any  term  by  the  exponent  of  the 
power  of  X  which  occurs  in  tliat  term,  and  divide  the  product  by  the 
number  of  the  term,  we  obtain  the  numerical  coefficient  of  the  next 
term. 

7.  The  r'*  term  will  be 


n(n  -l)(n-2)  ■■■  (n  -  r  -  2)  ^„_t^_i,^,_i 
1.2.3-  (r-~  1) 
(2) 

^  n(n  -  l)(n  -  2)  .-.(«-  r  +  2)  ^n-r+ij^r-i 
1  •2-3...  (r-1) 

The  numerical  coefficient  of  this  term  has  the  form  of  a  fraction  whose 
denominator  is  a  product  of  /•  —  1  factors,  namely,  the  product  of  the 
first  r  —  1  integers.  The  numerator  is  also  a  product  of  r  —  1  factors, 
namely  the  r  —  1  integers  n,  n  —  \,  •••  n  —  (/•  —  2)  which  are  obtained  by 
subtracting  in  order  0,  1,  2  •••?•  —  2  from  n. 

8.  The  last  or  (n  +  l)""  term  is  equal  to  h".  But  it  will  also  be  in- 
cluded under  formula  (2)  if  there  we  put  ?•  =  n  -f-  1,  provided  that  we 
make  the  agreement  that  the  symbol  ./"  shall  be  regarded  as  equivalent 
to  unity,  that  is,  x"  =  1. 


Akt.  88]  THE   BINOMIAL   THEOREM  143 

All  of  these  laws  are  contained  in  the  formula 

(x  +  A )"  =  i'"  +  -x"-^h  +  ^^^~^)a;"-2A2  +  ... 
1  1-2 

(^3^)     I  *  n(n  -  1 ) (n  -  2)  ■  •  •  (n  -  r  +  3)  ^„_,.^3^^_g 

t^Cn-l)(n-2)...0.-r  +  2)    ^._,,,^  ^ 

1  .2.8...  (r-2)(r-l)  ' 

and  the  truth  of  tliis  formula,  which  is  called  the  binomial 
theorem,  is  what  we  wish  to  prove. 

So  far,  we  only  know  that  the  formula  is  correct  for 
w=l,  2,  3,  4,  5.  We  wish  to  show  that  it  is  correct  for  all 
values  of  the  positive  integer  n.  In  order  to  do  this  we 
prove  the  following  lemma. 

If  the  binomial  theorem  (3)  is  correct  for  n  =  k,  it  ivill  also 
he  correct  for  n  =  k  +  \. 

Proof.  The  hypothesis  of  the  lemma  is  that  formula  (3) 
holds  for  n  =  k  ;   that  is,  that  the  formula 

{X  +  hy  =  x'  +  ^.r^-i/i  +  ^^1=_1)  2:^-2/^2  +  ... 

(-4 ^  .  K^-1)  •••  (^-r  +  3)    ,._,+27 ,-2 

^  ^  1-2-3...  (r-2) 

1  .2.3...  (r-1)      '^ 

is  actually  correct. 

Let  us  see  what  the  value  of  (x  +  hy^^  must  be  on  this 
hypothesis.  We  can  find  (x  +  A)*+i  from  {x  +  hy  by 
multiplying  the  latter  by  a;  +  A.     Consequently,  if  we  multi- 

*  This  is  the  (?•—  l)th  term,  and  is  obtained  from  (2)  if  we  replace  r  by  /•  —  1. 
+  This  is  the  rth  term  as  given  by  (2). 


144  INTEGRAL   RATIONAL   FUNCTIONS 

ply  both  members  of  (4)  by  x  +  A,  we  shall  find 


[Art. 


(•^) 


+ 


1  .  2.  a...  (r-l) 


1  1  .  2  .  3  ...  (r-2) 

+  ...  +  A'+i. 

Let  us  combine  those  terms  of  (5)  which  contain  the  same 
power  of  X  as  a  factor.  The  total  coefficient  of  t^Ii  will  be 
k  -\-\.     The  total  coefficient  of  x^~^W-  will  be 

k{k-\)      k  ^kfk-  1      ^^k  k  -\+2^k{k  +  \~) 
1.2  1      iV     2  y      i  2  1.2 

^(^-  +  l)(/^4-1-l) 
1-2 

The  total  coefficient  of  x^~''^Ve~'^  will  be  the  sum  of  the 
two  coefficients  of  a:^-'"+'-^7i'-i  in  (5).  But  the  first  of  these 
coefficients  may  be  written 

kjk-l)  •■■  jk-r  +  o)  k-r  +  2 
1.2...(r-2) 


r-1 


so  that  their  sum  is  equal  to 

k(J<:  —  1)  •••  (^  — r +  3)  Vk—r  +  2. 
1.2  .3  ...  (r-2)      L    r-1 


+  1 


^^(^-1)  ...  (A;-r  +  3)  ^  +  1 
1  .  2  .3'...  (r-  2)        r-1 


^(^+  t)(A;+1  -l)(A:+l-2)...  (A;+l-r  +  2) 
1.2.  3  ...  (r-2)(r-l) 

and  this  is  precisely  tlie  value  which  we  should  obtain  for 
this  coefficient  if  we  were  to  make  use  of  formula  (3)  for  the 
case  n  =  k  ->[- 1. 


Akt.  88]  THE   BINOMIAL   THEOREM  145 

Thus  we  have  actually  proved  that  if  the  rth  term  of  the 
expansion  of  (a:  +  hy  is  c^iven  by  fornri'ula  (2)  for  n  =  k,  then 
the  rth  term  in  (x -{■  A)'-+^  will  be  given  by  this  same  formula 
for  n  =  k-lt-^-  Since  the  rth  term  represents  ani/  term  of 
the  expansion,  we  liave  actuall}''  proved  our  lemin:i  :  if  the 
binomial  formula  is  correct  for  n  =  k,  it  will  also  he  correct  for 
n  =  k-\-'^. 

But  we  know  that  the  binomial  formula  is  correct  for 
n=  1,  2,  3,  4,  5.  Oar  lemma  allows  us  to  conclude,  without 
actual  test,  that  it  will  also  be  true  for  n  =  G.  A  second 
application  of  the  lemma  shows  the  formula  to  be  true  for 
w  =  7,  and  so  on,  for  all  positive  integral  values  of  7i. 

Formula  (2)  gives  the  rth  term  of  the  expansion  of 
(x  +  A)"-  The  formula  for  the  (r  +  l)th  term,  which  may  be 
called  the  rth  term  after  the  firsts  is  a  little  easier  to  re- 
member ;   it  is  equal  to 

^\  1.2.3...r 


EXERCISE  XXXVI 

Write  out  the  following  espausious  by  use  of  the  formula : 

1.  {x  +  hy.  7.    (2a-3/>)e.  12.    (1  +  J^) 

2.  {X  -  hy. 

3.  {a  +  hy. 

4.  {a -by.  '      \      ■   xJ  ,^_     /^^l 


7. 

(2  a  -  '^ly. 

8. 

(4x-5//)5. 

9. 

hi)' 

10. 

ii  +  \y- 

13.       1  + 


,!)■ 


5.  (-a  +  l>y. 

6.  (a'  +  l/^y.  11.    (1+i)'-  15-    (^-^0" 

16.  Use  the  binomial  theorem  to  compute  lOP. 
Hint.     Put  101  =  100  +  1. 

Use  the  binomial  theorem  to  compute  the  following  powers. 

17.  1026. 

18.  996. 

19.  (1.1)^*'  to  four  .significant  figures. 

20.  (1.01)1"°  to  four  significant  figures. 

21.  Find  the  eleventh  term  in  the  expansion  of  (x  +h)^'. 
Hint.     Use  the  formula  (2),  Art.  88,  with  n  -  17,  r  =  11. 


146  INTEGRAL   RATIONAL   FUNCTIONS  [Art.  89 

22.  Find  the  fourth  term  of  (a  -  4  by^. 

23.  Find  the  fifth  terra  of  (3x  -  2  >jy^ 

24.  Find  the  sixth  term  of  (  x  +  -  j    . 


25.    Find   the  rth  term  of 


(■^^r■ 


26.  Find  the  middle  term  of  {x  +  /0^°. 

27.  Find  the  two  middle  terms  of  (x  +  /i)"- 

28.  Equations  (1),  Art.  88,  seem  to  indicate  that,  in  the  expansion  of 
(x  +  hy,  the  numerical  coefficients  equidistant  from  the  ends  are  equal. 
Prove  that  this  is  so  by  making  use  of  formula  (2)  of  Art.  88. 

29.  Equations  (1),  Art.  88,  seem  to  indicate  that  the  middle  term 
when  n  is  even,  and  the  two  middle  terms,  when  n  is  odd,  have  the 
greatest  numerical  coefficients.     Prove  that  this  is  so. 

30.  Write  out  the  expansions  for  (a  +  ft  +  c)^  and  {a  A-  h  +  c)*. 
Hint.     Use  the  binomial  formula  with  x  =  a  A-  h,  and  h  =  c. 

89.   The  derivative  of  an  integral  rational  function.     It  is 

now  an  easy  matter  to  find  the  derivative  of  the  function 

(1)  fQx)  =  Ax^  +  Bx^~^  +  6V-2  +  ...^Lx  +  M. 
According  to  the  binomial  theorem  we  have 

/(^  +  ^0 

=  A{x+hy->rB(x->rhy-'^-\-CQx-\-hy-''-+--  +L{x  +  h)+M 

+  (7[.?:"-2^(?i-2):C"-3/i+  ..■]-!-  ...  +  L{x  +  K)+M, 

where  the  terms  which  contain  /i^  or  a  higher  power  of  h  as 
a  factor  have  not  been  written  down,  but  are  merely  indi- 
cated by  dots.  If  we  collect  those  terms  which  contain  no 
h  at  all,  and  those  which  contain  h  as  a  factor,  we  find 

f(x  +  h)  =  Ax""  +  5a;"-i  +  Ox^-"^  +  ■.-  +Lx  +  M 

(2)  +  h[nAx--^  +  (w  -  l)5.c"-2  +  (n  -  2)  Ox^-^  +  •..  +  i] 
+  terms  each  of  which  contains  h^  as  a  factor. 


Akt.  90]      DERIVATIVE   OF   AX   INTEGRAL   FUNCTION    147 

From  (1)  and  (2)  we  find,  by  subtraction  and  division 

bv  h, 

f(x  +  h)-f(x) 

h 

^^^  =  wAx»-i  +  (w  -  l)5a:"-2  +  (m  -  2)  (7a;"-3  +  •  •  •  +  i^ 

+  terms  each  of  which  has  A  as  a  factor. 

The  derivative  of  f(x)  is  the  limit  which  (3)  approaches 
when  h  approaches  zero  (Art.  87).  That  part  of  (3) 
which  has  been  written  out  will  remain  unchanged  as  h 
approaches  zero,  because  it  contains  no  h.  All  of  the  other 
terms  of  (3)  will  have  the  limit  zero  since  each  has  A  as  a 
factor.  Therefore,  the  derivative  of  the  function  /(a:),  de- 
fined by  (1),  is 

(4)  fix)  =  nAx^-^  +  (n  -  l}Bx"-^  +  (»i  -  2)  Cx'^-^  +  ■■•  +  L. 

Observe  that  the  law  according  to  which  f  (x)  is  obtained 
from  f(jjc)  is  very  simple.  Each  term  of  /(^)  produces  a 
corresponding  term  of  f'(x)  by  applying  the  following  rule: 
Multiply  any  term  of  f(x)  by  the  exponent  of  the  power  of  x 
which  occurs  in  it,  and  afterward  reduce  this  exponent  by  unity. 

This  rule  may  even  be  regarded  as  applying  to  the  last 
terra  M  of  /(a:),  which  at  first  sight  seems  to  be  an  excep- 
tion, inasmuch  as  it  produces  no  corresponding  terra  in  f  {x). 
For  we  may  think  of  M  as  the  coefficient  of  aP  =  1.  (See 
Art.  88  No.  8  of  the  fine  print.)  If  we  do,  the  above  rule 
will  give  0  as  the  term  oi  f  (x^  which  corresponds  to  the 
term  M\nf(x). 

EXERCISE   XXXVII 

1.  Solve  a  second  time  the  examples  of  Exercise  XXXV,  making  use 
of  formula  (4),  Art.  89,  for  the  purpose  of  computing  the  derivatives. 

2.  Prove  directly  that  the  derivative  of  a  constant  is  equal  to  zero. 

90.  Derivatives  of  higher  order.  We  now  know  how  to 
find  the  derivative  f'(x)  of  any  integral  rational  function 
f(x).  \i  f(x')  is  of  degree  n,f'{x^  is  an  integral  rational 
function  of  degree  n~l.     The  derivative  of /'(a;),  denoted 


148  INTEGRAL   RATIONAL   FUNCTIONS  [Art.  91 

by/"  (a:),  will  therefore  be  an  integral  rational  function  of 
degree  w— 2,  and  is  called  the  second  derivative  of /(a;). 
We  may  continue  in  this  way.  Clearly  the  kth  derivative 
of  /(a;)  will  be  an  integral  rational  function  of  degree 
71  —  k.  In  particular  the  nth  derivative  of /^a;)  will  be  a 
constant,  that  is,  it  will  contain  no  x.  The  {n  +  l)th  deriva- 
tive and  all  derivatives  of  order  higher  than  this  will  be  equal 
to  zero.      (See  Ex.  2,  Exercise  XXXVII.) 

EXERCISE  XXXVIll 

Compute  the  derivatives  of  higher  order  for  all  of  the  functions  men- 
tioned in  Exercise  XXXV. 

91.  Taylor's  expansion.  By  making  use  of  the  higher 
derivatives  we  can  now  write  the  expansion  of  f{x+  K)  in  a 
very  simple  form.  We  shall  give  the  details  of  the  proof 
only  for  the  case  where /(a;)  is  of  the  third  order,  so  that 

(1)  /  (x)  =  Ax^  +  Bx^  +  Gx  +  i), 

but  the  same  method  would  apply  to  an  integral  rational 
function  of  ;iny  degree. 

Let/'  {x),f"  (x),f"'  (a;),  etc.,  represent  the  first,  second, 
third  derivatives,  etc.,  of/  (x).     Then  we  find  from  (1), 

fix)  =  3Ar2  +  -lBx+  0, 
fix)  ^6  Ax  +  2  B, 
^"^^    .  /'"(a:)  =  6A 

/4^(a:)=/'5>(a:)=  ••.  =0. 

Again,  we  find  from  (1) 

fix  +  h)  =  A<ix  +  hf+  B{x  +  hy  +  C(x  +  A)  +  B 

or,  if  we  expand  each  of  these  terms  by  the  binomial 
theorem,  and  arrange  according  to  powers  of  h, 

QV)  f(x+h)  =  Ax^-\-Bx^+  Cx+B  +  h{?j  Ax^  +  2Bx+  (7) 
+  h\d  Ax  +  B)+  h^A. 

The  sum  of  the  first  four  terms  of  the  right  member  is 
/(a;)  ;  the  coefficient  of  h,  according  to  (2),  is  equal  to /'(a;); 
the  coefficient  of  h^  is  equal  to  one  half  of /"(a:)  ;  the  coeffi- 


Art.  92]     DIMINISHING   ROOTS   OF   AN   EQUATION  149 

cient  of  A^  is  equal  to  one  sixth  of /'"(a:).      We  may  there- 
fore rewrite  (3)  as  follows: 

(4) /(:r  +  /0=/(a:)  +  ^/'(.r)+ ^/''(.r)-f  ^-|-^/'''(a:). 

By  applying  the  same  method  to  a  function  of  higher 
degree  we  obtain  the  following  result  : 

Iff(x)  is  an  integral  rational  function  of  the  nth  degree,  we 
may  express  fix  +  A),  as  a  sum  of  terms  arranged  according  to 
ascending  powers  of  h,  in  the  form 

(5)  f(x+  h)  =fCx)  +  P'(x)  +  j^/'C^)  +  YTtr/"^''^ 
+  -  +  r-^r^3 — /"^(^)' 

1  •  J,  '  o  •■•  n 
where  f"\x)  denotes  the  nth  derivative  off(x'). 

This  formula  is  known  as  Taylor  s  expansion  iov  f{x  +  h). 


EXERCISE  XXXIX 

1.  Givenf(x)  =  i3^-2x-'  +  7  x-o.     Fmd/(x  +  h),  f{l  +  h),f(2  +  h). 

2.  Giveuf(x)=x*-Sx^  +  x^-l.   Fmdf(x+h),f(-l+h),/(-2  +  h). 

3.  Work  out  the  detailed  proof  of  formula  (5),  Art.  91,  for  the  cases 

n  =  i,  and  n  —  5.  ' 

92.  Diminishing  the  roots  of  an  equation.  We  have  seen 
in  Art.  82  how  approximate  values  may  be  obtained  for  the 
real  roots  of  an  equation 

(1)  f{x)  =  Ax''  +  Bx"^^  +  -.-  +  Lx+  M=  0, 

by  inspecting  the  graph  of  the  function  y  =  f(^x'). 

Let  us  suppose  that  a  is  an  approximation  to  a  root  of  (1) 
obtained  in  this  way,  and  let  the  exact  value  of  the  root  be 
X  =  a  -{-  h,  so  that  h  is  the  unknown  correction  (positive  or 
negative)  wliich  must  be  added  to  a  to  insure  that  a  +  h 
maybe  exactly  a  root  of  the  eciuation  (IV  We  shall  then 
liave 

(2)  f(a  +  h)=0. 


150  INTEGRAL   RATIONAL  FUNCTIONS  [Art.  92 

Thus,  the  correction  h  must  satisfy  the  equation  (2).  But 
this  equation  is  obtained  from  (1)  by  writing 

(3)  X  =  a  -\-  h,  ov  h  =  X  —  a. 

Consequently  every  root  h  of  (2)  will  be  less  by  a  than 
a  corresponding  root  x  of  (1).  We  have  the  following 
result. 

If  a  is  an  approximation  to  a  real  root  of  the  equation 
f^x^  =  0,  the  correction  h  (^positive  or  negative'^,  tvhich  makes 
a  -\-  h  exactly  a  root  of  the  equation,  ivill  itself  satisfy  the  equa- 
tion f  (a  +  A)=  0,  all  of  whose  roots  are  less  hy  exactly  a  than 
the  corresponding  roots  off(x')=  0. 

It  is  important  therefore  to  be  able  to  determine  the  equa- 
tion in  ^,  ^^     ,   7\      A 

whose  roots  differ  from  those  of  /(.»)  =  0  by  a.  The  prob- 
lem of  obtaining  this  equation  is  known  as  diminishing  the 
roots  of  the  given  equation  hy  a. 

But  we  can  solve  this  problem  at  once.  We  need  merely 
to  put  re  =  a  in  formula  (5)  of  Art.  91.     This  gives 

(4)  /(a  +  70  =/(a)  4-  \f\a~)  +  ^/"(a)  + 

1  .  2...  n 

as  the  equation,  with  h  as  its  unknown,  all  of  lohose  roots  are  less 
hy  exactly  a  than  those  of  the  equation  /(a')  =  0.  Both  equa- 
tions are  of  the  same  order. 

The  problem  of  increasing  the  roots  of  an  equation  by 
a  is  included  in  the  above,  if  we  admit  negative  values 
of  a. 

Since /(x)  is  a  known  function  of  x,f'(x^,f"(x'),  etc.,  may 
be  found  by  Arts.  89  and  90.  Since  a  is  a  given  number 
f(cL),f'(^a'),  etc.,  may  be  calculated,  and  thus  the  coefficients 
of  equation  (4)  may  be  determined.  It  only  remains  to  find 
rules  enabling  us  to  perform  this  calculation  conveniently. 


Art.  9:1]      ARRANGEMENT   OF   THE   CALCULATION  151 

93.  Arrangement  of  the  calculation.  In  order  to  accom- 
plish this,  we  return  to  tlie  calculation  explained  in  Art. 
82.  We  there  found  the  following  method  for  calculating 
/(a)  when 

(1)  f(x)  =  Ai^  +  33^+  Cx  +  D. 

We  put 

(2)  E=Aa  + B.  F  =  Ea+ C=Aa^  +  Ba+ C, 


and  arranged  the  work  as  follows  : 

ABC 

B 

(S)                                   Aa       Ea 

Fa 

E         F       /(a) 

Let  us  now  add  Aa  to  E  and  call  the  sum  (r,  so  that 

(3)  Aa  +  E=Aa  +  {Aa  +  B')  =  2Aa  +  B=a. 

If  we  multipl}^  (r  by  a  and  add  the  product  to  F^  we  find, 
making  use  of  (2)  and  (3), 

(4)  Ga  +  F=2Aa'^  +  Ba-^{Aa'^+Ba+ C^ 

=  ^Aa^  +  2Ba-\-C. 

But  this  is  equal  to /'(a)  on  account  of  (2),  Art.  91. 
This  calculation  is  actually  carried  out  by  adding  two  new 
lines  to  the  scheme  (*S'),  which  will  then  read  as  follows: 


A 

B 

CD             \_a 

Aa 

Ea       Fa 

{S') 

E 

F        f(a) 

Aa 

aa 

a 

/'(«) 

Let  us  now  add  Aa 

to  a. 

We  find,  from  (3), 

(5)                     Aa  +  a 

=  3  Aa 

+  ^  =  A/» 

152  INTEGRAL   RATIONAL   FUNCTIONS  [Art.  93 

on  account  of  the  second  equation  of  (2),  Art.  91.  Finally, 
on  account  of  the  third  equation  of  system  (2),  Art.  91,  we 
have 

0  L  '  1  •  6 

We  now  complete  the  scheme  (/^S")  by  adding  another  line 
which  embodies  the  calculations  just  indicated.     This  gives 


(^'0 


A 

B 

Aa 

C  B  \a 
Ea       Fa 

E 

Aa 

F        f{a)* 

aa 

G 
Aa 

j\a)* 

A* 

1 
1.2 

fXar 

The  quantities  marked  with  an  asterisk  will  he  the  coefficients 
of  the  equation  for  A,  whose  roots  are  less  by  exactly  a  than  those 
of  the  given  equation. 

For,  on  account  of  (4),  Art.  91,  this  equation  may  be  written 

1.2-3  1-2  1  ^  ^ 

The  method  of  calculation  described  here  may  be  applied  to  integral 
rational  functions  of  any  degree  and  is  not  confined  to  cubics.  The  ex- 
planation has  been  given  for  the  case  of  cubics  merely  for  purposes  of 
brevity. 

EXERCISE  XL 
1.    Diminish  the  roots  of  3  .r^  —  4  x^  +  7  x  —  1  =  0  by  2. 
Solution.     The  calculation  (S")  here  becomes: 
3    -    4+    7-    1L2 


6+    4  +  22 

+  2+11  +  21* 
6+  16 

+  8  +  27* 
6 

3*+  14* 


Art.  91]  MULTIPLICATION   OF   THE   ROOTS  153 

Therefore  the  equation,  each  of  whose  roots  is  less  than  tlie  correspond- 
ing root  of  the  given  equation  by  2  units,  is  :^  h^  +  14  h-  +  27  h  +  21  =  0. 

2.  Increase  the  roots  of  3  x'^  —  4  x-  +  7  .t  —  1  =  0  by  2. 
Hint.     Diminish  the  roots  by  —  2. 

In  each  of  the  following  examples  diminish  the  roots  by  the  number 
indicated  in  the  parenthesis. 

3.  2  X*  -  :]  x-^  +  4  X  -  5  =  0.     (2) 

4.  j-3  +  :}  X  -  20  =  0.     (2) 

5.  x8  +  0  x2  +  15  X  -  6  =  0.     (.3) 

6.  2  x^  +  16  x3  +  45 x'-^  +  5(3  x  +  2:5  =  0.     (-2) 

7.  x3  -  0  x2  +  10  X  +  6  =  0.      (o) 

8.  5  X*  -  6  x3  +  8  X  -  04  =  0.     (15) 

9.  Work  out  the  details  of  Art.  93  for  the  case  of  a  quartic  equation, 

Ax*  +  Bx^  +  Cx2  +  Dx  +  E  =  0. 

94.   Multiplication  of  the  roots  of  an  equation  by  m.     Let 

X  be  a  root  of  the  equation 

(1)  Ax''  +  Bx""-^  -\ \-  Lx  +  M=  0, 

and  let  us  attempt  to  find  a  second  e<iuation,  with  the  un- 
known x\  whose  roots  are  just  m  times  as  great  as  those 
of  (1).      We  shall  then  have 

(2)  x'  =  mx,  x  =  —  • 

m 

If  we  substitute  this  value  of  x  in  (1),  we  find  the  equation 

a(^\  +  ^f^Y"'  + ...  +  //i:!  +  it/= 0 

\mj  \m/  m 

for  x' .  But  this  may  be  simplified  by  multiplying  both 
members  by  tw",  giving 

(3)  Ax'""  +  mBx'""'^  +  m^Cx'"''^  +  •••  +  m'^-'^Lx'  +  w"il^=  0. 

We  obtain  the  following  rule.  To  find  an  equation  whose 
roots  shall  he  m  times  as  (/reat  as  those  of  a  given  equation^  ive 
proceed  as  follows.     Arrange  the  terms  of  the  given  equation 


154  INTEGRAL   RATIONAL   FUNCTIONS  [Art.  95 

according  to  descending  poivers  of  the  vnknoiim  quantity,  a  zero 
coefficient  being  supplied  for  each  of  the  powers  of  the  unknown 
ivhich  does  not  occur  in  the  equation.  Multiply  the  coefficients 
in  order  by  1,  m,  ?n^,  wi^,  •••  m".  Tlie  resulting  products  are 
the  coefficients  of  the  desired  equation. 

While  this  rule  is  convenient  and  easy  to  remember,  it  is 
still  simpler  in  any  given  case  to  actually  make  the  substitu- 
tion (2)  which  led  to  the  rule. 

95.  Changing  the  sign  of  the  roots.  An  important  special 
case  of  the  transformation  just  explained  arises  when  m  =  —  1. 
In  that  case  we  have  x'  =  —  x  and  the  roots  of  the  two 
equations 

(1)  Ax'^  +  Bx""-^  +  Gx""-^  +  i>:r"-3  +  . . .  =  0 

and 

(2)  Ax'^  -  Bx'--^  +  CV"-2  -  Dx'""'^  +  . . .  =  0 

are  so  related  that,  to  every  root  of  (1)  there  corresponds  a 
root  of  (2)  of  the  same  magnitude  but  of  opposite  sign. 

This  transformation  is  very  useful  when  tve  are  attempting 
to  calculate  a  negative  root  of  (1),  since  it  enables  us  to  calculate 
instead  the  corresponding  positive  root  of  (^2). 

EXERCISE  XLI 

In  Examples  1-4  find  the  equation  whose  roots  are  obtained  from  those 
of  the  given  equation  by  multiplication  with  the  number  indicated  in 
parenthesis. 

1.  X3+  2x2  +  1  X+3  =  0.      (^(;)  3.    .7-''- 2x3+ 7x2-1  =  0.      (2) 

2.  2  x3  +  7  X  -  1  =  0.     (;3)  4.    x3  _  i  x-  -  |  =  0.     (.5) 

5.  From  each  of  the  equations  in  Examples  1-4  find  another  one 
whose  roots  shall  be  numerically  equal  to  those  of  the  given  equation, 
but  opposite  in  sign. 

Use  the  transformation  of  Art.  94  to  obtain  from  each  of  the  follow- 
ing equations  another  one  all  of  whose  coefficients  shall  be  integers,  and 
for  which  the  coefficient  of  the  highest  power  of  the  unknown  shall  be 
equal  to  unity.  Use  the  smallest  value  of  ?/*  which  will  accomplish  the 
purpose. 


Aht.  !)()]    CONTINUITY   OF   INTEGRAL  FUNCTIONS  155 

6:    ,r^-  I  .c2  +  1  X  +  ',  =  0.  8.    L>  ./•*  +  3  x^  +  2  x  +  3  =  0. 

7.    3  ./:3  +  X-  +  7=0.  9.    .r3  -  J  x2+  ^  x^  +  *|  =  0. 

10.  It  may  happen  that  the  transformation  of  Art.  95  does  not  change 
the  equation  at  all ;  that  is,  the  coeHicients  of  (1)  and  (2),  Art.  95,  may 
be  identical.  Under  what  circumstances  will  this  happen,  and  what  can 
you  conclude  concerning  the  roots  of  such  an  equation? 

96.  Continuity  of  integral  rational  functions.  In  calculat- 
ing a  real  root  of  an  e(jiuitioii  of  the  form 

(1)  f(x)  =  Ax"  4-  i^x"-!  +  -'.  +  Lx  +  31=  0, 

we  may  proceed  as  follows.     We  first  make  a  graph  of  the 
function 

(2)  ^=/(^)- 

The  real  roots  of  (1)  will  be  the  abscissas  of  the  points  in 
which  the  graph  crosses  the  a;-axis.  Suppose  that,  in  mak- 
ing the  calculations  required  for  drawing  this  graph,  we 
happen  to  strike  a  value  of  x  which 
makes  f{x)  exactly  equal  to  zero.  Then, 
of  course,  this  value  of  a;  is  a  root  of  the 
equation.  Ordinarily,  however,  this  will 
not  occur ;  usually  none  of  the  values  of 
X,  say  a,  b,  c,  d,  etc.,  which  are  used  in 
these  calculations  will  be  roots  of  the 
equation ;  so  that  the  values  of  /(«),  f(J>'), 
/(c),  etc.,  will,  all  of  them,  be  different  from  zero.  Suppose 
/(a)  <  0,  and  f{h)  >  0,  a  case  illustrated  in  Fig.  43,  where 

MF  =f(a-)  <  0,  NQ  =f{b)  >  0. 

The  figure  leads  us  to  conclude  that  there  will  be  at  least 
one  point,  between  M  and  iV,  where  the  curve  crosses  the 
a;-axis.  If  a^j  is  the  abscissa  of  such  a  point,  a-^,  which  lies  be- 
tween a  and  b,  will  be  a  root  of  the  equation  (1). 

Although  this  argument  sounds  very  plausible  we  have, 
so  far,  no  guarantee  for  its  correctness.  If  the  graph  of  the 
function  /(x)  is  indeed  an  unbroken  (continuous)  curve  as 
indicated  in  Fig.  43,  the  conclusion  will  be  correct.     But  if 


156  INTEGRAL   RATIONAL    FUNCTIONS  [Akt.  96 

the  graph  were  broken  ((liscontiiuious)  as  in  Fig.  44;  the 
conclusion  might  be  erroneous.  We  shall  see  later  that  such 
graphs  as  that  of  Fig.  44  are  not  at  all 
uncommon,  but  they  never  present  them- 
\  selves  wlien  f  {x)  is  an  integral  function. 

M         1^ ^    It   can   be    shown   that    the    graph    of    an 

I  *  integral   rational   function   is  always  con- 

^  tinuous. 

Fig.  44 

In  order  that  we  may  express  these  con- 
ditions somewhat  more  precisely,  we  formulate  the  following 
definition : 

A  function  f  (x)  is  said  to  be  continuous  in  the  neighborhood  of 
a  particular  value  of  x^  say  x  =  p^  if  the  difference 

f(p+h)-f(p) 

approaches  zero  for  its  limit,  whenever  h  approaches  zero  in  any 
manner,  through  positive  or  negative  values,  continuously,  or  by 
jumps. 

If  the  function  is  continuous  in  the  neighborhood  of  all 
values  of  x  which  lie  between  a  and  b,  the  function  is  said  to 
be  continuous  in  the  interval  from  a  to  b. 

The  graph  of  a  continuous  function  will  be  an  unbroken 
(continuous)  curve,  such  as  is  illustrated  in  Fig.  43.  This 
figure  suggests  the  following  theorem. 

I.  If  the  function  f  (x)  is  continuous  in  the  interval  from  a  to 
b,  and  if  the  values  off(^a^  andf(^l)}  are  opposite  in  sign,  there 
will  exist  at  least  one  value  of  x,  between  a  and  b,  for  which  the 
function  f  (^x^  will  become  equal  to  zero. 

The  argument  which  we  attempted  to  make,  at  the  be- 
ginning of  this  article,  will  therefore  be  justified  if  the 
following  statement  is  correct. 

II.  An  integral  rational  function  of  x  is  continuous  in  any 
finite  interval. 

Theorems  I  and  II  are  both  correct.  We  shall  not  however 
attempt  to  prove  them  now,  since  the  proofs  are  somewhat 


Art.  97]     NEWTON'S   METHOD   OF   APPROXIMATION         157 

abstract.  The  discussion  just  given,  and  the  experience 
which  we  are  gradually  gaining  in  the  drawing  of  graphs, 
will  serve  to  make  both  theorems  seem  plausible. 

97.  Newton's  method  of  approximation.  Let  a;  be  a  root  of 
the  equation  f{x)=0  ;  let  a  l)e  an  approximate  value  of  this 
root,  obtained  perhaps  by  inspection  of  the  graph,  and  let  us 
put  re  =  a  +  A,  so  that  h  is  the  correction  whicli  must  be 
added  to  a  to  obtain  the  true  value  of  the  root.  According  to 
Art.  92,  tliis  unknown  correction  h  will  be  a  root  of  the 
equation 

(1)  /(a)  +/'(«)/.+  l/"(a)A2+  ...  =0. 

Now,  if  a  is  a  fairly  good  approximation  to  x,  h  will  be  a 
comparatively  small  fraction  of  a  so  that  the  ratios  h^/a\ 
h^/a^,  etc.,  will  be  small  as  compared  with  h/a.  In  most  cases, 
therefore,  we  may  expect  to  get  a  good  approximation  to  the 
correction  h  by  neglecting  the  terms  which  involve  A^,  Jfi, 
etc.,  in  (1).  If  we  do  this,  (1)  reduces  to  an  equation  of  the 
first  degree  for  h,  namely 

/(a)  +  /'Ca)A=0, 
which  gives 

(2)  A  =  _ZO!l. 

If  A  is  computed  by  this  formula,  a  +  A  =  a^  will  usually  be 
closer  to  the  true  value  of  x  than  was  a ;  but,  in  general,  a^ 
will  still  not  be  the  exact  value  of  a:,  but  only  an  approxima- 
tion. If  now  we  repeat  this  process,  using  a^  instead  of  a, 
we  may  compute  the  quantities 

'''  =  -/(S'  ''2  =  ''.  +  ^.- 

and  usually  a^  will  be  a  still  better  approximation  to  the 
root.  In  most  cases  this  process  of  approximation,  if  repeated 
often  enough,  will  enable  us  to  obtain  the  value  of  x  with  any 
desired  degree  of  accuracy.  The  process  is  known  as  Newton's 
method. 


158  INTEGRAL   RATIONAL   FUNCTIONS     [Arts.  98,  99 

98.   Geometric   significance   of   Newton's   method.      Let   A 
(Fig.  45)  be  the  point  of  the  graph  of  y=if(x)  which  cor- 
responds to  the  value  of  x, 

x  =  a  =  OM. 
Ihen 

MA^fCa), 

and  the  slope  of  the  tangent  ^7^  is  equal 
to  /'(a).     (See  Art.  87.)     Let  H  be  the 
point  in   which  the   tangent   crosses   the 
a:-axis,  and  let  a  +  7i  be  the  abscissa  of  IT.     Then  the  coordi- 
nates of  if  will  be  (a -I- A,  0),  while  those  of  A  are  (a,  /(«))• 
Therefore  (see  Art.  53),  the  slope  of  J. 2^  will  be  equal  to 

/(«)-Q    ^    /(«), 

a  —  (a  +  A)  /* 

Since  the  same  slope  is  also  equal  to  /'(a),  as  we  have  seen 
above,  we  must  have 

wlience  h  =  —  ■ 


/'(«) 


But  this  is  precisely  the  value  of  h  given  by  equation  (2)  of 
Art.  97.  Therefore  Newton's  method  of  approximation  consists 
in  replacing  the  curve  ABO  by  the  straight  line  AT^  which  is 
tangent  to  the  curve  at  the  point  A  whose  abscissa  is  equal  to  a. 

The  approximate  value  of  x  obtained  by  a  single  applica- 
tion of  Newton's  method  is  the  abscissa  a^  of  the  point  H 
(Fig.  45).  A  second  application  of  the  method  would  con- 
sist in  replacing  that  part  of  the  curve  between  A^  and  B  by 
the  tangent  at  Ay  The  figure  shows  how  very  nearly  the 
intersection  of  this  tangent  with  the  aj-axis  would  coincide 
with  B. 

99.  The  method  of  false  position  (Regula  falsi).  A  second 
method  of  approximation,  also  suggested  by  geometry,  is  as 
follows.    If /(a)  and/ (6)  are  opposite  in  sign,  we  know  that 


Art.  99]  THE   METHOD   OF   FALSE   POSITION 


159 


Fig.  46 


there  is  at  least  one  root  of  the  equation  /(.c)  =  0  between 
a  and  b.     In  Fig.  46  we  have 

a  =  031  /(a)  =  3IA  <  0, 
b=ON,  f(h)  =  NB>0, 

and  the  root  of /(a')=0,  which  lies  be- 
tween a  and  b,  is  the  abscissa  of  the  point 
R.  li  A  and  B  are  close  together,  we  may 
regard  the  straight  line  AB  as  a  reasonably  close  approxi- 
mation to  the  curve  ABB,  and  the  point  S,  in  which  AB 
intersects  the  2:-axis,  as  a  fairly  good  approximation  to  the 
point  R.  The  abscissa  of  jS  will  therefore  be  an  approxima- 
tion to  that  root  of  the  equation  /(a;)  =  0  which  corresponds 
to  tlie  point  B.  We  shall  now  show  how  to  calculate  this 
approximate  value  of  the  root. 
The  slope  of  the  line  AB  is 

/W-/(«) 


(1) 


m  = 


(Art.  53) 


since  the  coordinates  of  A  and  B  are  (a, /(a))  and  (^,/(i)) 
respectively.  The  equation  of  a  straight  line  which  passes 
through  the  point  A  (a, /(a)),  and  whose  slope  is  equal  to 

771   IS 

Therefore 
(2) 


..  .      f(b^-f(a). 


b—  a 


a) 


is  the  equation  of  the  line  AB. 

To  find  the  abscissa  of  the  point  S  in  wliich  AB  intersects 
the  aj-axis,  we  must  put  ^  =  0  in  (2)  and  compute  the  corre- 
sponding value  of  X,  which  we  shall  call  x-^.     Thus  we  find 

-/(«)=/W-/('')(.,-a). 
b  —  a 


whence 

or 

(3) 


X-,  —  a  — 


x,  =  a 


-f(a)(b-a) 

f(a)(b-a^ 
'fib)-f(a^ 


160 


INTEGRAL   RATIONAL   FUNCTIONS         [Art.  100 


The  value  of  x^  given  hy  (3)  is  the  abscissa  of  the  point  S^ 
and  will  in  general  furnish  a  close  approximation  to  the 
desired  root  if  a  and  b  are  close  enough  together. 

By  repeating  this  process,  using  x^  and  6,  or  a  and  a:^,  instead 
of  a  and  6,  we  may  obtain  a  second  approximation,  and  so  on. 

This  method  may  be  regarded  as  the  geometrical  equiva- 
lent of  the  following  arithmetical  argument.  If  a  and  b  are 
close  together,  and  if  x  is  between  a  and  5,  the  change  in 
the  function /(a;)  will  be  approximately  proportional  to  the 
change  in  the  variable  x.  Now  as  x  changes  from  a  to  a:^, 
f(x)  will  change  from  /(a)  to  f{x^);  as  x  changes  from  a  to 
6,  f(x)  will  change  from  /(a)  to  f(h^.  Therefore  we  shall 
have  approximately  : 


(4) 


f(h)-fia)       b-a 


If  x^  is  approximately  a  root  of  the  equation  f{x)  =  0,  we 

shall  have  very  nearly /(a-j)  =  0.     If  we  substitute /(a;j)=  0 

in  (4),  we  find 

-fjo-^      ^x^  —  a 

which  again  gives  the  value  (3)  when  solved  for  x^ 

One  may  usually  obtain  a  very 
close  approximation  by  combining 
Newton's  method  with  the  method 
of  false  position.  For  the  true 
value  of  the  root  usually  lies  be- 
r  tween  the  two  approximate  values 
obtained  by  these  different  meth- 
ods. Thus,  in  Fig.  47,  the  point 
M  lies  between  the  points  H  and  aS'. 

100.   An  example  of  Newton's  method.     We  wish  to  find  a 
real  root  of  the  equation 

(1)  f(x')  =  x^  +  J^-Sx^-  x-4  =  0. 

We  find         /(I)  =- 6, /(2)  =+ 6. 


Art.  100]     AN   EXAMPLE   OF   NEWTON'S  METHOD 


161 


Consequently  (Art.  96,  Theorems  I  and  II),  there  is  a  real 
root  between  x  =  1  and  x  =  2.     Therefore  we  put 

a:  =  1  4-  2/,  or  7/  =  3-  —  1, 

where  ?/  is  a  positive  proper  fraction  ;  ?/  is  the  correction 
which  must  be  added  to  1  so  as  to  make  1  +  y  a  root  of  (1); 
it  is  the  quantity  which  was  denoted  by  h  in  Art.  97. 
According  to  Art.  92,  t/  will  be  a  root  of  the  equation  ob- 
tained from  (1)  by  diminishing  its  roots  by  1.  We  perform 
the  calculation  by  synthetic  division  : 


1      -3 
1      +2 


-4|1 
_  2 


2 
1 

-1     -2 
+  3     +2 

-6* 

3 
1 

+  2 
+  4 

0* 

4 

1 

+  6* 

(2) 


1*       5* 

Thus,  the  correction  y  will  be  a  root  (between  0  and  1)  of 
the  equation 

(3)  g(l/)  =  y'  +  f>f  +  ^f  +  0-i/-  6  =  0. 

It  frequently  happens  that  Newton's  method  fails  to  furnish 
a  good  approximation  at  the  first  stage  of  the  calculation 
even  if  it  works  satisfactorily  at  the  later  stages.  It  fails 
entirely  at  the  first  stage  of  tliis  example  because  the  co- 
efficient of  J/  in  (3)  is  equal  to  zero.  But  the  method  of 
false  position  (Art.  99)  may  be  used  to  advantage.  If  we 
use  formula  (3)  of  Art.  99,  putting  a  =  1,  6  =  2,  we  find 

'  6 -(-6)  -2 

thus  suggesting  1.5  as  an  approximate  value  for  x,  or  0.5  as 
an  approximate  value  for  ^. 


162 


INTEGRAL   RATIONAL   FUNCTIONS         [Ai:t.  100 


To  locate  the  root  a   little    more    precisely  we    calculate 
^(0.5)  by  synthetic  division.     This  gives 

15  6  0  -6  |Q.5 

(4)  0.5         2.75        4.375  2.1875 


5.5 


8.75 


4.375 


-3.8125 


We  have  thus  found  ^(0.5)  =  —  3.8125.     Since  we  have 

i/CO)  =/(l)  =  -  <3,   ^(1)  =/(2)  =  +  (3, 

the    true    value    of   the  correction   i/  must  lie  between  0.5 
and  1.0. 

Let  us  try  y  =  0.8  next.     Sjaithetic  division  gives 

15  6  0  -6  |0.8 

(5)  0.8        4.64         8.512         +6.8096 


5.8       10.64         8.512 


+  0.8096 


so  that  ^(0.8)  =+ 0.8096.  Since  ^(0.5)  = -3.8125,  the 
root  of  (3)  for  which  we  are  looking  lies  between  0.5  and 
0.8.  Since  the  value  of  ^(0.8)  is  small,  it  is  apparent  that 
the  approximation  0.8  for  i/  is  fairly  close  and  we  shall  there- 
fore continue  our  calculation  based  upon  the  approximations 
1.8  for  a:  or  0.8  for  y,  by  putting 

(6)  ^  =  0.8  +  2. 

To  determine  the  equation  which  the  correction  z  must 
satisfy,  we  must  therefore  diminish  the  roots  of  (3)  by  0.8. 
We  have  already  performed  a  part  of  this  calculation  in  (5). 
We  now  complete  this  transformation. 


5 

0.8 


4.64 


0 
8.512 


-6  |0. 

6.8096 


5.8 
0.8 


10.64 

5.28 


8.512 
12.736 


(7) 


+  0.8096* 


6.6 
0.8 
7.4" 
0.8 
1*       8.2* 


15.92 
5.92 
21.84* 


21.248* 


Art.  100]     AN   EXAMPLE  OF   NEWTON'S  METHOD 


163 


Thus  z  is  a  root  of  the  equation 
(8)       h(z)  =  s4-f  8.2  z3  +  21.84  s2  +  21.248  z+  0.8096  =  0. 

Newton's  method,  which  consists  essentially  in  neglecting 
z^,  z3,  s*  in  the  above  equation,  leads  us  to  regard 

CQ\  0.8006 

^  ^  21.248 

as  an  approximate  value  of  z.  It  is  not  safe  to  assume  that 
this  value  will  be  correct  to  more  than  its  first  significant 
figure.  We  therefore  use  as  our  approximate  value  of  z, 
suggested  by  (9),  z  =  —  0.04,  giving  j/  =  0.8  —  0.04  =  0.76 
and  a;  =  1.76  as  the  approximate  value  of  a  root  of  (1) 
probably  correct  to  the  second  decimal  place. 

We  now  continue  this  process  so  as  to  get  a  still  closer 
approximation.     Since  (8)  has  a  root  approximately  equal 
to  —  0.04,  we  put 
(10)  z  =  -0.04  +  ^ 

that  is,  we  diminish  the  roots  of  (8)  by  —  0.04  (or  increase 
them  by  +  0.04).  This  gives  rise  to  the  following  calcu- 
lation: 


0.8096 


0.04 


-0.04 

-  0.3264 

-0.860544 

-0.81549824 

8.16 
-0.04 

21.5136 

-  0.3248 

20.387456 
-0.847552 

-  0.00589824* 

8.12 
-0.04 

21.1888 
-0.3232 

19.539904* 

(10) 

8.08 
-0.04 

20.8656* 

1*       8.04* 

This  leads  to  the  approximate  value 
-  0.00589824 


t 


19.539904 


=  0.000302, 


where  only  the  last  figure  is  uncertain.     For  it  can  be  shown 
that,  if  the  quotient  fia)/f'{a}  begins  ivitli  k  zeros  when  ex- 


164  INTEGRAL   RATIONAL   FUNCTIONS         [Art.  101 

pressed  as  a  decimal  fraction,  then  the  best  approximation  is 
obtained  by  carrying  out  the  divisioyi  to  '2  k  decimal  places. 

Thus  we  have  found 

a;  =1.76  +  ^  =  1.760302, 

the  last  figure  only  being  in  doubt. 

When  the  student  has  become  familiar  with  the  process, 
the  discussion  given  in  the  text  of  this  article  may  be 
omitted,  and  the  actual  calculation  to  be  exhibited  will  con- 
sist only  of  the  numbers  in  the  schemes  (2),  (7),  (10). 

Since  the  problem  of  finding  the  nth.  root  of  a  given  num- 
ber a  is  equivalent  to  the  solution  of  the  equation 

a:"  —  a  =  0, 

Newton's  method  is  also  applicable  to  this  problem.  In  fact, 
even  in  the  case  of  a  square  root  and  cube  root,  the  calcu- 
lation by  Newton's  method  is  far  more  convenient  than  the 
method  taught  in  elementary  algebra. 

101.  Horner's  method.*  Horner's  method  is  very  similar 
to  Newton's  method.  The  principal  difference  consists  in 
the  fact  that,  in  Horner's  method,  negative  corrections  are 
avoided.  In  our  example  in  Art.  100  we  found  1.8  as  an 
approximation  to  the  root  and  then  proceeded  to  improve  our 
knowledge  of  this  root  by  starting  out  in  our  calculations 
with  this  approximate  value.  If  we  had  been  using  Horner's 
method,  we  should  not  have  used  1.8  as  an  approximation  ; 
for  1.8  is  greater  than  the  required  root,  as  is  brought  out 
by  our  calculation,  since  the  correction  turns  out  to  be 
negative.  Consequently  by  Horner's  method,  at  this  stage, 
we  should  proceed  as  follows.  As  soon  as  we  have  dis- 
covered that  the  root  is  less  than  1.8,  we  try  the  value  1.7 
for  X.  We  should  then  find  that  1.7  is  the  correct  approxi- 
mation to  use  for  x  in  tlie  application  of  Horner's  method  ; 
since  the  root  actually  lies  between  1.7  and  1.8,  the  first  two 
significant  figures  of  the  root  are  1.7  and  the  correction 
to  this  will  be  positive.     We  then  proceed  as  in  Newton's 

*  W.  G.  Horner,  London  Philosophical  Transactions,  1819. 


Arts.  102,  103]  NEGATIVE   ROOTS  165 

method,  using  1.7  as  our  approximate  value  of  x  instead  of 
the  value  1.8  whicli  we  actually  used. 

In  this  example  1.8  was  actually  a  closer  approximation 
to  the  root  than  1.7.  Consequently  in  this  case  Horner's 
method  would  not  be  quite  as  advantageous  as  Newton's. 
In  general,  we  may  expect  to  obtain  a  result,  correct  to  a 
given  number  of  significant  figures,  more  rapidly  by  New- 
ton's than  by  Horner's  method.  F'or  Newton's  method 
allows  us  the  privilege  of  using  that  one  of  two  numbers 
between  which  the  required  root  is  known  to  lie,  whicli  is 
the  closer  approximation.  The  only  advantage  of  Horner's 
method  is  that  all  of  the  corrections  are  positive. 

102.  Abbreviated  calculation.  It  often  happens,  especiall}"- 
in  problems  of  applied  mathematics,  that  only  a  certain  num- 
ber of  significant  places  are  required.  In  fact,  in  most  such 
problems,  the  coefficients  of  the  equation  are  themselves  not 
known  with  absolute  accuracy.  Their  values  depend  upon 
certain  measurements,  and  owing  to  the  uncertainties  of 
such  measurements,  their  values  are  only  known  with  an 
accuracy  of  a  certain  number  of  decimal  places.  The  exact 
application  of  Newton's  or  Horner's  method,  however,  usu- 
ally introduces  decimal  places  far  beyond  these.  But  these 
additional  decimal  places  add  nothing  to  the  accuracy  of 
the  desired  result  from  the  practical  point  of  view.  Much 
labor  may  therefore  be  avoided  by  not  calculating  these 
higher  decimal  places  at  all.  The  following  is  a  good  prac- 
tical rule.  If  in  a  practical  problem  a  root  of  an  equation  is 
to  be  found  correct  to  k  decimal  places,  abbreviate  all  the  num- 
bers ivhich  occur  in  the  calculation  to  k  -\- 1  decimal  places,  and 
abbreviate  the  final  result  to  k  decimal  places. 

103.  Negative  roots.  Negative  roots  may  be  obtained  by 
the  same  process  as  positive  roots.  It  is  usually  preferable, 
however,  to  first  make  the  transformation  of  Art.  95,  thus 
reducing  the  problem  of  computing  a  negative  root  of  the 
given  equation  to  that  of  computing  a  positive  root  of  the 
transformed  equation. 


166  INTEGRAL   RATIONAL   FUNCTIONS     [Arts.  104,  105 

104.  Computation  of  more  than  one  real  root.  After  one 
root  a  of  an  equation  has  been  determined,  the  problem  of 
computing  a  second  real  root  (if  th^e  is  a  second)  may,  of 
course,  be  treated  in  the  same  way.  It  is  simpler,  however, 
to  first  divide  the  left  member  of  the  equation  hy  x  —  a^  and 
to  continue  the  calculation  with  the  equation  of  lower  degree 
obtained  in  this  way.  This  equation  is  called  the  depressed 
equation.  Moreover,  if  Newton's  or  Horner's  method  was 
used  for  calculating  the  first  root,  the  coefficients  of  the 
depressed  equation  have  already  been  determined  in  the  last 
synthetic  division,  so  that  everything  is  prepared  for  the 
determination  of  the  next  real  root. 

EXERCISE  XLII 

1.  Find  to  two  decimal  places  the  root  of  x^  +  ?>  x  —  20  =  0  which 
lies  between  2  and  3. 

2.  Find  to  three  decimal  places  the  root  ofa;^  —  2x  —  5  =  0  which 
lies  between  2  and  3. 

3.  Find  to  four  decimal  j^lfices  the  root  of  x^  +  5  .r  —  7  =  0  which 
lies  between  1  and  2. 

4.  Find  to  two  decimal  places  the  root  of  a;^  +  3x-  —  2  a;  —  5  =  0 
which  lies  between  1  and  2. 

5.  Find  a  positive  root  of  x^  =  63  correct  to  two  decimal  places. 

6.  Find  the  real  fifth  root  of  37  correct  to  two  decimal  places. 

7.  Find  to  two  decimal  places  tJie  root  of  x^  —  '2x'^  —  'd  x  +  Q  —  0 
which  lies  between  —  1  and  —  2. 

8.  If  the  coefficients  of  the  highest  and  lowest  powers  of  x  which 
occur  in/(a;)  have  opposite  signs,  the  equation  f(x)=  0  has  at  least  one 
positive  root.     Prove  this  statement. 

Hint.  Discuss  separately  the  cases :  I,  when  the  equation  has  no 
zero  roots;  II,  when  it  has  zero  roots.  In  case  I,  observe  that/(.r)  will 
assume  values  opposite  in  sign  for  x  =  0  and  for  x  sufficiently  large  and 
positive.     Then  apply  Art.  96. 

105.  Upper  limit  for  the  positive  roots  of  an  equation.  Let 
us  turn  our  attention  once  more  to  the  example  of  Art.  100, 
and  more  specifically  to  the  last  line  of  the  synthetic  division 
(5)  of  that  article.     The  numbers  which  occur  in  this  line 


Art.  10.-.]     IPPEU    LTMTT    FOR   THE   POSITIVE   ROOTS       1G7 

are  all  positive,  and  the  last  one,  +  0.8096,  is  the  value  of 
^(7/)  for  2/  =  0.8,  or  of  f(x)  for  x  =  1.8.  From  the  method 
of  calculation  it  is  evident  that  any  value  of  x  greater  than 
1.8  would  only  serve  to*  increase  the  value  of  f(x^.  Conse- 
quently no  root  of  the  equation  can  be  as  great  as  1.8,  and 
we  may  say  that  1.8  is  an  upper  limit  for  positive  roots  of 
this  equation. 

But  what  was  the  significance  of  the  numbers  in  the  last 
line  of  the  synthetic  division  (5)  of  Art.  100  ?  According 
to  Art.  86  this  synthetic  division  teaches  us  that  the  quotient 
obtained  in  dividing  .r*  +  a:^  —  3 a;^  —  2.  —  4  by  a;  — 1.8  is 
a;3+ 5. 8a'^  +  10.64a;  +  8.512,  and  that  the  remainder  is 
+  0.8096. 

Putting  these  two  things  together,  our  example  suggests 
the  following  theorem.  Let  f(x^  be  an  integral  rational 
function,  in  which  the  coefficient  of  the  highest  poiver  of  x  is 
positive.  Let  a  he  a  positive  number,  let  Q(jc)  be  the  quotient, 
and  let  R  be  the  remainder  obtained  when  we  divide 
/(a:)  by  X  —  a.  If  R  is  positive  and  if  none  of  the  coefficients 
of  Q(^x)  are  negative,  then  a  is  an  upper  limit  for  the  positive 
roots  of  the  equation  f  (x)  =  0.  Tliat  is,  no  positive  root  of  this 
equation  can  be  as  great  as  a. 

The  proof  is  very  simple.     We  have 

f(x~)  =  Q(x}  (x  -  a)  +  R.  (Art.  85) 

Since  R  is  positive  and  Q{x}  has  no  negative  coefificients 
the  right  member  will  be  positive  for  all  values  of  x  as 
great  as,  or  greater  than,  a.  That  is,  no  number  as  great  as, 
or  greater  than,  a  can  be  a  root  of  the  equation /(.r)=  0. 

Of  course  the  value  of  R  and  the  coefficients  of  Q{x^  are 
best  obtained  by  synthetic  division.  We  may  therefore  re- 
state our  theorem  in  the  following  form  which  is  especially 
well  adapted  for  practical  application. 

If'  in  the  synthetic  division  of  f(x^  by  x  —  a,  a  being  positive, 
none  of  the  results  in  the  third  line  are  negative,  tJien  a  is  an 
upper  limit  for  the  positive  roots  of  the  equation  f  {x^=-  0. 


168  INTEGRAL  RATIONAL  FUNCTIONS         [Art.  106 

In  applying  this  critei'ion  it  is  again  presupposed  that  the  highest 
power  of  X  in/(x-)  has  a  positive  coefficient.  If  this  coefficient  should 
happen  to  be  negative,  change  the  sign  of  all  of  the  coefficients  before 
proceeding  farther. 

The  knowledge  which  results  in  a  specific  case  from  this 
statement  will  save  us  from  wasting  time  in  searching  for 
roots  of  the  equation.  For  we  need  not  test  an}'^  values  of  x 
as  large  as  the  upper  limit.  Thus  in  our  example,  if  we 
wish  to  find  a  second  real  root  of  the  equation  we  know  that 
it  cannot  be  as  large  as  1.8. 

For  other  methods  of  obtaining  upper  limits  see  Dickson's 
Elementary  Theory  of  Equations,  page  57. 

Since  all  questions  concerning  negative  roots  may  be  con- 
verted into  questions  concerning  positive  roots  of  another 
equation,  by  means  of  the  transformation  of  Art.  95,  we  need 
not  expressly  formulate  the  corresponding  criterion  for  the 
lower  limit  of  the  negative  roots  of  an  equation. 

106.  Descartes's  rule  of  signs.  It  is  evident  that  an 
equation 

(1)  /  (a:)  =  Ax^  +  5:r«-i  +  .-.  +Lx  +  M=^ 

can  have  no  positive  roots  if  all  of  the  coefficients  have  the 
same  sign,  that  is,  if  there  are  no  variations  of  sign  among 
the  coefBcients.  For  the  value  of  f(x)  will  obviously  be 
positive  for  all  positive  values  of  x  if  all  of  its  coefficients 
are  positive  ;  it  will  be  negative  for  all  positive  values  of  x 
if  all  of  the  coefficients  are  negative.  In  neither  case  can  it 
become  equal  to  zero  for  a  positive  value  of  x. 

The  question  arises  whether  there  may  not  be  a  more 
general  connection  between  the  number  of  variations  of  sign 
among  the  coefficients,  and  the  number  of  positive  roots. 
That  such  a  relation  exists  was  discovered  by  Descaktes. 

Let  us,  as  usual,  arrange  the  terms  of  the  equation  accord- 
ing to  descending  powers  of  the  unknown  quantity,  and  let 

(2)  A,  B,  C,  i), ...  i,  M 

be  the  coefficients  of  a;",  x"~^,  •••  x,  1  in  this  order.  If  A 
were  negative,  we  might   consider   the  equivalent  equation 


Art.  106]  DESCARTES'S   RULE   OF   SIGNS  169 

—  j(^x)=  0,  whose  A  would  then  be  positive.  We  may 
therefore  assume  that  A  is  positive.  We  now  inspect  the 
coefficients  (*2),  beginning  with  A^  in  the  order  written.  If 
they  are  all  positive,  we  say  there  is  no  variation.  If,  on 
the  other  hand,  we  observe  k  changes  of  sign  among  the 
coefficients  (2),  as  we  read  them  in  the  order  written,  we 
say  that  there  are  k  variations.  Zero  coefficients  may  be 
omitted  in  counting  variations. 

Descartes's  rule  says  that  an  equation  with  k  variations  has 
at  most  k  positive  roots. 

To  prove  this  rule  we  proceed  as  follows.  Let  a:^,  x^,  •••  x^ 
be  all  of  the  positive  roots  of  the  equation  /(a:)=  U.  Then, 
according  to  the  factor  theorem  (Art.  84),  x  —  x^,  x  —  x^^  •••, 
x  —  x^  are  factors  Q)if{x^.  Consequently  the  division  of/(x) 
by  {x  —  x^{x  —  Xc,^  •••  {x—x^)  will  be  exact,  and  the  (quo- 
tient g{x)  will  be  an  integral  rational  function  of  degree 
n  —r  such  that  the  equation  g{x)  =  0  has  no  positive  roots. 
The  polynomial  g{pc)  may  have  some  variations  or  else  it 
may  have  none  ;  that  is  a  matter  about  which  we  profess 
ignorance.  We  put  this  ignorance  into  evidence  by  letting 
Fo  represent  the  number  of  variations  in  ^(:c),  it  being  under- 
stood that  Vf;^  may  be  equal  to  zero  or  some  positive  integer. 

Let  us  now  write  down  the  signs  of  the  coefficients  of 
^(a:),  as  follows  : 

(8)        g{x)  =  +•••  + +  •••  + +  •••  +, 

where  the  dots  indicate  any  number  of  terms,  whose  coeffi- 
cients have  the  same  sign  as  those  between  which  the}^  are 
placed.  As  has  been  stated  already,  V^  denotes  the  number 
of  variations  in  g{x). 

We  now  attempt  to  determine  the  number  of  variations  in 
the  product  (^x  —  x^  )g(^^  where  x-^  is  a  positive  number.  If  we 
perform  the  multiplication  in  the  usual  manner,  however,  writ- 
ing down  only  the  signs  of  the  various  partial  products,  we  find 

(a:  -  x^)g(x)  = 

+  +  •••  4--  -  +   +-  -+  •••  + 

-  -+  +-  -+  +-  - 

+  ±  •••  ±-±  •••  ±  +  ±  •••  ±-±  •••  ±  +  ±  •••  ±-, 


170  INTEGRAL  RATIONAL  FUNCTIONS         [Art.  106 

where  the  ambiguous  sign  ±  indicates  those  terms  of  the 
product  concerning  the  sign  of  whose  coefficients  we  can  say- 
nothing  definite.  Since  in  any  particular  case  some  of  these 
ambiguous  signs  may  be  replaced  by  +  and  others  by  — ,  it 
is  evident  that  in  some  cases  the  product  {x  —  x^)g(x}  may 
have  man?/  more  variations  than  ^(2;).  But  in  all  cases  this 
product  will  contain  at  least  one  more  variation  than  ^(x). 
For  we  shall  certainly  not  be  overestimating  the  number  of 
variations  in  the  product  if  we  replace  all  of  the  ambiguous 
signs  of  a  group  by  the  sign  of  the  term  which  just  precedes  the 
group.     If  we  do  this  the  signs  of  the  product  are  as  follows : 

+  +•••+-- +  •••  + +•••+-. 

But  this  arrangement  of  signs  is  the  same  as  in  g(^x}  except- 
ing the  last,  which  gives  rise  to  an  extra  variation. 

Thus,  if  Fj  denotes  the  number  of  variations  in  the  prod- 
uct (x  —  x{)g(^x^,  Fj  is  at  least  greater  by  one  unit  than  V^. 
That  is  Ti^Fo  +  l- 

Let  us  now  multiply  (x  —  x{)g{x')  by  a:  —  x^,  where  x^ 
is  a  second  positive  root  of  the  original  equation  /(a;)  =  0. 
Let  V^  denote  the  number  of  variations  in  the  product 
{x  —  x^{x—  Xc^^g(x).     By  the  same  argument  we  find 

and  therefore  V^  ^  Vq  +  2. 

Let  us  proceed  in  this  way  until  we  have  multiplied  gix)  by 
the  product  of  x  —  Xy,  x  —  x^^  •"  x  —  x^.  The  complete  prod- 
uct is  equal  to/(a;).  If  Vr  denotes  the  number  of  variations 
in  /(a;),  we  have  therefore 

or  r  ^  Vr  -  To- 

Since   Vq  may  be  equal  to  zero  or  a  positive  integer,  we 
shall  certainly  have 
(4)  r  <  Vr. 


Art.  106]  DESCARTES'S  RULE  OF   SIGNS  171 

Now  the  equation  f  {x)  —  0  has  precisely  r  positive  roots 
and  Vr  variations.  Therefore  the  inequality  (4)  states  in 
symbols  precisely  what  Descartes's  rule  states  in  words, 
namely : 

The  number  of  positive  roots  of  an  equation  f  (a:)  =  0  cannot 
exceed  the  number  of  variations  among  the  coefficients  of  the 
equation. 

It  slioiild  be  noted  that  Descartes's  rule  does  not  state  that  an  equation 
has  as  many  positive  roots  as  it  has  variations.  It  merely  says  that  the 
equation  can  have  no  more  positive  roots  than  variations.  It  may  have 
that  nuiny  positive  roots  or  it  may  have  fewer.  Thus,  in  the  simplest 
case  when  there  are  no  variations,  we  may  say  at  once  tliat  there  are  no 
positive  roots.  But  an  equation  may  have  variations  and  still  have  no 
positive  roots.     Thus  the  quadratic  equation 

a;2  _  X  +  1  =  0 

has  two  variations  but  no  positive  root,  both  of  its  roots  being  imaginary 
as  may  be  verified  by  solving  the  equation. 

We  may  state  Descartes's  rule  in  a  more  definite  form  on 
the  basis  of  the  following  remarks.  The  equation  g(x)  =  0 
was  assumed  to  have  no  positive  roots.  We  made  Use  of  this 
assumption  tacitly  when  we  wrote  down  the  signs  of  g(x)  in 
(3),  by  using  a  group  of  -|-  signs  for  the  last  group  in  (3). 
For,  if  the  last  group  of  signs  in  (3)  had  consisted  of  —  signs, 
the  equation  g(x^=  0  would  have  to  possess  at  least  one  pos- 
itive root.  (See  Ex.  8,  Exercise  XLII.)  Since  the  first  and 
last  group  of  signs  in  gCx)  both  consist  of  +  signs,  g(x)  can 
contain  only  an  even  number  of  variations,  if  it  has  any  vari- 
ations at  all.  Thus  Vq  is  either  equal  to  zero  or  an  even 
integer. 

Again,  when  we  were  estimating  the  smallest  number  of 
variations  which  the  product  (x  —  x{)g(x)  could  possibly 
have,  we  replaced  the  first  group  of  signs  in  the  product, 
-\-  ±±  •••  ±,  which  was  followed  by  a  —  sign,  by  the  group 
-I-  +  -I-  ...  -(-  followed  by  a  —  sign.  Any  one  of  the  terms 
of  this  group  except  the  first  might,  however,  have  a  negative 
coefficient.  But  if  we  change  any  one  or  several  of  the  -f- 
signs  in  the  group  -h  +  +  •••  +»  except  the  first,  which  is 


172  INTEGRAL   RATIONAL   FUNCTIONS         [Art.  107 

not  doubtful,  to  — ,  the  number  of  variations  in  the  group 
will  change  by  an  even  number  or  not  at  all,  never  by 
an  odd  number.  Similarly  for  each  of  tlie  other  groups. 
Consequentl}"  the  true  number  of  variations  of  tlie  product 
(x  —  x^g(x)  can  differ  from  Vq-{-  1  only  by  an  even  number. 
If  we  combine  these  remarks  with  our  former  argument, 
we  obtain  the  following  more  precise  statement  of  Descartes's 
rule  : 

The  equation  f  (2)  =  0  has  either  as  maiiy  positive  roots  as 
there  are  variations  among  the  coefficients  off(x)^  or  else  fewer 
hy  an  even  number. 

Descartes's  rule  may  also  be  used  in  discussing  negative 
roots.  The  transformation  of  Art.  95  enables  us  at  once  to 
make  the  following  statement : 

An  equation  f(x')  =  0  has  either  as  many  negative  roots  as 
there  are  variations  among  the  coefficients  of  f(^—  x),  or  else 
fewer  by  an  even  number. 

EXERCISE    XLIII 

In  the  following  examples  find  the  niaxiniuni  number  of  positive  and 
negative  roots  by  applying  Descartes's  rule. 

1.  a;3  +  5  X  -  7  =  0. 

2.  x3  +  2  X-  +  8  =  0. 

3.  x3  +  1  =  0. 

4.  x"  +  1  =  0. 

5.  x"  -  1  =  0. 

11.  Instead  of  counting  the  variations  among  the  coefficients  as  in  Art. 
106  (2)  we  may  count  the  number  of  permanences,  that  is,  the  number  of 
times  that  the  coefficients  of  /(x)  written  in  the  order  (2)  of  Art.  106, 
fail  to  change  sign.  A  complete  equation  is  an  equation  of  the  form 
f(x)  =  0  in  which  no  term  of  /(x)  has  a  zero  coefficient.  Prove  that 
a  complete  equation  has  no  more  negative  roots  than  the  number  of 
permanences  among  its  coefficients. 

107.    Maxima  and  minima  of  an  integral  rational  function. 

We  have  learned  how  to  compute  tlie  slope  of  the  straight 
line  which  is  tangent  to  the  graph  of  y  =/(2;)  at  the  point 


6. 

X*  +  3-3-3  X-  +  X  -  3 

=  0. 

7. 

.H   +   j;-2   -1=0. 

8. 

2  x3  +  7  x2  -  5  =  0. 

9. 
.0. 

x^  +  x«  -1  =  0. 

xs  +  x3  -  7  X  +  1  =  0. 

Art.  107]  MAXIMA   AND   MINIMA  173 

(x,  y).  Tilt'  slope  of  this  tangent  is  equal  to  the  derivative, 
f'(x),  of  f{x).  (See  Art.  87.)  We  have  also  seen  (in 
Art.  53)  that  a  line  slopes  upward  from  left  to  right  or 
downward  from  left  to  right  according  as  its  slope  is  positive 
or  negative.  Consequently  the  course  of  a  curve  y  =/(a;) 
will  he  upward  from  left  to  right  in  the  neighhorhoo'1  of  a  point 
whose  abscissa  makes  f  {x}  positive,  and  downward  from  left 
to  right  near  a  point  for  tohichf^pc)  is  negative. 

Jn  other  words,  the  function /(a;)  increases  with  increasing 
X  when  its  derivative  /'(.c)  is  positive;  it  decreases  with  in- 
creasing X  \\\\enf'(x)  is  negative. 

A  maximum  is  a  point  on  a  curve  which  has  a  greater 
ordinate  than  any  other  point  in  its  immediate  neighbor- 
hood. It  is,  therefore,  a  point  at  which 
the  curve  changes  from  an  upward  to 
a  downward  course  if  we  think  of  the 
curve  as  being  described  from  left  to 
right.  (See  the  point  marked  A  in  Fig. 
48.) 

Similarly  at  a  minimum   (such    as  ^, 
Fig.  48)  the  curve  changes  from  a  downward  to  an  upward 
course. 

Consequently,  if  jo  is  a  particular  value  of  x  which  corre- 
sponds either  to  a  maximum  or  minimum  of  the  graph  of 
y  =  f(x},  the  derivative /'(a;)  must  change  its  sign  as  the 
variable  x  passes  through  the  value  jo.  If/ (2;)  is  an  integral 
rational  function, /'(a;)  is  also  an  integral  rational  function 
and  therefore  a  continuous  function.  (See  Art.  96.)  Con- 
sequently/'(a;)  can  change  sign  only  by  passing  through  zero. 
We  draw  the  following  conclusion : 

The  abscissas  of  the  maxima  or  minima  of  an  integral  rational 
function  f{x)  are  included  among  those  values  of  xivhich  make 
the  derivative  f  (x^  equal  to  zero. 

But  not  all  roots  of  the  ecjuation  f'{x)  =  0  need  to  corre- 
spond to  maxima  or  minima  of /(a;).  In  fact  the  equation 
/'(a;)  =  0  really  only  means  that  the  slope  of  the  tangent  is 


174  INTEGRAL   RATIONAL   FUNCTIONS  [Art.  107 

equal  to  zero,  that  is,  that  the  tangent  at  such  a  point  is 
parallel  to  the  a^-axis.  And  this  may  take  place  at  a  point, 
such  as  (7,  Fig.  48,  which  is  neither  a  maximum  nor  a 
minimum. 

In  order  to  distinguish  between  these  cases,  we  may  pro- 
ceed as  follows.  Suppose  that  x  =  p  is  a  root  of  the  equation 
f  '(x)  =  0,  which  is  obtained  by  equating  to  zero  the  derivative 
oi  f(x).  Let  h  be  a  very  small  positive  number,  so  that  —h 
is  negative.  We  examine  the  signs  of  f'(p—1i)  and 
f\p  +  ^)»  ^^^  obtain  the  following  criteria  : 

If  f'ip-Ji)>^,  /'(f)  =  ^'  /'(io  +  ^)<0,   x  =  p  gives  a 

maximum  of/(:r). 
If/'(^-A)<0, /'(jt>)  =  0, /'(^  +  /0>0,    x=:p    gives   a 

minimum  of /"(a;). 
If   /'(^-A)>0,    /'(jt>)=0,    /'(jt>  +  70>0,    x  =  p    gives 

neither  a  maximum  nor  minimum. 
If/'(^-A)<0,   /'(p)=0,    /'(;,+  A)<0,    x  =  p    gives 

neither  a  maximum  nor  minimum. 

The  determination  of  the  maxima  and  minima  of  a  function 
/(a;)  is  often  a  matter  of  great  practical  importance.  Most 
of  the  proljlems  of  Engineering  are  questions  of  tliis  kind, 
since  the  engineer  should  attempt  to  make  his  constructions 
serve  their  purpose  with  a  maximum  of  efficiency  for  a  given 
outlay  in  money  and  time.  From  our  present  point  of  view, 
it  is  evident  that  a  knowledge  of  the  maxima  and  minima 
of  a  function  is  bound  to  be  of  great  assistance  in  studying 
its  graph. 

EXERCISE    XLIV 

Discuss  the  functions  given  in  Examples  1  to  5  for  maxima  and 
minima. 

1.  ?/  =  x2  +  2  :c  -  3.     ■  3.    ?/  =  3  2-2  +  2  X  -  1. 

2.  y/  =  —  2  .r-  +  7  X  +  5.  4.    y  =:  —  5  j:-  -f  2  a;  +  2. 

5.  y  =  ax'^  +  hx  +  c.  Compare  your  result  as  obtained  by  the  use  of 
the  derivative  with  the  result  previously  obtained  in  Art.  66. 

6.  Find  the  maxima  and  minima  of  ?/  =  x^  —  3  x^  +  2  x.  Plot  the 
curve  and  find  the  roots  of  the  equation  x^  —  3  x^  +  2  x  =  0. 


Art8.  108,  109] 


MULTIPLE   ROOTS 


176 


7.  Examine  the  function  ,y  =  x^  —  10 a^  +  30  for  maxima  and  minima 
and  plot  tlie  curve. 

8.  Has  the  function  y  =  (x  —  oy  a  maximum  or  minimum?  Prove 
the  correctness  of  your  answer. 

9.  Examine  the  function  y  —  x^  —  (i  x-  +  10  for  maxima  and 
minima. 

10.  Examine  the  function  y  =  x(x'-  —  1)  for  maxima  and  minima. 

11.  A  box  open  at  the  top  is  to  be  made  from  a  square  piece  of  tin,  the 
length  of  one  side  of  the  square  being  a  inches.  It  is  proposed  to  do  this 
by  cutting  equal  squares  out  of  the  four  corners  and  then  bending  up  the 
tin  so  as  to  form  tlie  sides  of  the  box.  AVhat  should  be  the  size  of  the 
squares  cut  out  of  the  corners  so  that  the  box  may  have  the  largest  pos- 
sible volume  ? 

12.  The  strength  of  a  beam  is  approximately  proportional  to  its  breadth 
and  the  square  of  its  depth.  What  are  the  dimensions  of  the  strongest 
beam  that  can  be  cut  out  of  a  circular  log  whose  diameter  is  d  inches? 

108.  Rolle's  theorem.  Let  J.ifil/2^3^  (^'igs.  49  and 
50)  be  the  graph  of  an  integral  rational  function  1/  =  fipf)^ 
which  crosses  the 
rr-axis  at  the  points 
A  and  B  whose  ab- 
scissas are  equal  to  a 
and  h  respectively. 
Then  we  have 

/(a)=/(i)  =  0. 

The  figures  show  that  there  will  be  at  least  one  maximum, 
or  at  least  one  minimum,  between  a  and  h.  Since  the  deriva- 
tive f  {x)  will  be  equal  to  zero  at  such  a  maximum  or  mini- 
mum, these  figures  suggest  the  following  theorem,  which  is 
known  as  Rolle's  theorem. 

If  an  integral  ratiowtl  funetion  f(^x)  has  the  zeros  x=  a  and 
X  =  by  so  that  f{a~)  =/(/>)  =  0,  then  there  exists  at  least  one 
value  of  X,  between  a  and  b,  for  which  the  derivative  f'(x) 
becoynes  equal  to  zero. 

109.  Multiple  roots.  We  shall  not  attempt  to  give  a 
formal  proof  of  Rolle's  theorem,  but  proceed  immediately  to 
make  an  important  application  of  it.     Let  us  think  of  the 


176  INTEGRAL   RATIONAL   FUNCTIONS         [Art.  109 

a:-axis  in  Fig.  51  as  fixed,  but  let  us  think  of  the  curve  AMB 
as  being  gradually  lowered  from  its  original  position,  first 
to  A'M'B',  and  then  into  the  third  posi- 
tion shown  where  the  curve  is  tangent  to 
the  ic-axis  at  M".  During  this  process 
the  three  points  A,  M^  B  approach  each 
other,  and  finally  all  three  of  these  points 
coincide  with  each  other  at  M" .  We  see  therefore  that  a 
point  of  contact,  such  as  M"^  may  be  regarded  as  arising 
from  the  union  of  two  real  points  of  intersection. 

If  m"  is  the  abscissa  of  M'\  the  function /(a;),  of  which  the 
lower  curve  of  Fig.  51  is  the  graph,  will  of  course  have 
X  —  m"  as  a  factor,  since  /(a:)  becomes  equal  to  zero  for 
X  =  m".  But  f(x^  will  actually  contain  {x  —  m")^  as  a  fac- 
tor, since  each  of  the  two  factors  x  —  a  and  x  —  b  {a  and  b 
being  the  abscissas  of  A  and  B^  will  tend  toward  x  —  m"  as 
the  curve  AMB  is  lowered  into  its  final  position.  We 
express  this  fact  by  saying  that  m"  is  a  double  root  of  the 
equation /(a-)  =  0. 

According  to  Rolle's  theorem  there  is  a  root  of  the  equa- 
tion f'(x^  =  0  between  any  two  roots  of  /(.r)  =  0.  In  our 
case  tliis  root  of  f  (x)  =  0  clearly  coincides  with  m" .  This 
is  also  apparent  from  the  fact  that  the  x-axis  is  tangent  to 
the  graph  at  M" . 

The  same  conclusion  follows  if  m"  is  a  triple,  quadruple, 
or  multiple  root  of  any  degree  of  multiplicity.  Consequently 
we  obtain  the  following  theorem: 

Every  multiple  root  of f(x)  =  0  i%  also  a  root  off'{x)  =  0*. 

By  the  factor  theorem  then,  every  multiple  factor  of  ./"(a:) 
will  also  be  a  factor  of /'(a;).  We  may  therefore  decide 
whether  a  given  equation  has  multiple  roots  or  itot  as 
follows: 

We  determine  the  highest  common  factor  of  f(^x)  and  f  {x). 
If  this  highest  common  factor  does  not  contain  x,  the  equation 

*The  converse  of  this  theorem  is  not  true.  That  is,  not  every  root  of /'(.'•)  =0 
is  a  multiple  root  off{x)  =  0,  nor  indeed  necessarily  a  root  oi/{x)  =  0  at  all. 


Art.  no]      RATIONAL    KOOTS  OF   AN    EQUATION  177 

f(^z)  =  0  han  no  multiple  root.  If  the  highest  common  factor 
does  contain  a,",  the  multiple  roots  of  /(a-)  =  0  will  be  those  roots 
of  the  equation  ivhich  are  obtained  by  equatim/  to  zero  the  highest 
common  factor  of\f\u)  andf'{x^. 

The  process  of  finding  the  highest  common  factor  of  two 
polynomials,  such  as  f(x)  and  /'(a;),  is  pi-obably  familiar  to 
the  student  from  his  first  course  in  algebra.  It  is  essentially 
the  same  as  the  process  for  finding  the  greatest  common 
divisor  of  two  integers  (see  Art.  5)  and  is  justified  by  the 
same  kind  of  reasoning. 

EXERCISE  XLV 

Examine  the  following  eiiuations  for  multiple  roots.  Determine  the 
multiple  roots  if  there  are  any  ;  and  use  these  roots  for  the  purpose  of 
finding  an  equation  of  lower  degree  which  the  remaining  simple  roots 
will  have  to  satisfy.     Solve  the  etpiations  completely. 

1.  ^-3  -  7  z-  +  1<>  X  -  12  ^  0.  3.  .r*  +  6  xH  x^-2ix  +  16=  0. 

2.  X*  -  6  x2  -  8  2  -  ;}  =  0.  4.  x"  -  0  x3  +  10  x^  -  8  =  0. 

5.  x^  -  15  x8  +  10  x2  +  60  X  -  72  =  0. 

6.  x5  -3x*-  o  x3  +  13  x^  +  24  X  +  10  =  0. 

7.  Show  that  x^  +  9  x'^  +  2  x  —  48  =  0  has  no  multiple  root. 

8.  Show  that  x^  +  :>  ^x  +  r  =  0  has  a  multiple  root  if  and  only  if 
4  78  +  r-  =  0. 

110.  Rational  roots  of  an  equation  with  rational  coefficients. 
If  all  of  the  coefficients  of  an  equation  of  the  form 

(1)  Ax""  +  Bx"-'^+  •■'  +  Lx  +  M=0 

are  rational  numbers,  we  may  at  once  reduce  the  equation  to 
another  one  of  the  same  form  with  integers  as  coefficients. 
To  do  this,  it  suffices  to  multiply  both  members  of  the  equa- 
tion by  the  lowest  common  denominator  of  its  various  frac- 
tional coefficients.  Let  us  assume  therefore  that  all  of  the 
coefficients  of  (1)  are  integers. 

If  we  divide  both  members  of  (1)  l>y  A  we  obtain  the 
equation 

(2)  a:"-f  ^r«-M-  •••  +^^  +  '^=0, 
^  ^  A  A        A 


178  INTEGRAL   RATIONAL   FUNCTIONS         [Art.  110 

iu  which  the  coefficient  of  a;"  is  equal  to  unity,  but  in  which 
the  remaining  coefficients  will  not,  in  general,  be  integers. 
But  if  we  put 

(3)  x  =  ^,  1/  =  kx, 

y  must  satisfy  the  equation, 

y^      Btr^      ,,.LyM^^ 
k-     Ak''~^  Ak     A 

or 

(4)  y-  +  !%«-!  +  ...  +  |f-V+  ^^"  =  0, 

and  the  integer  k  may  always  be  chosen  in  such  a  way  that 
this  equation  for  y  shall  have  integral  coefficients.  (Com- 
pare with  Art.  94.)  In  fact  if  we  put  k  =  A,  the  coefficients 
of  (4)  will  certainly  be  integers,  but  often  a  value  of  k 
smaller  than  A  will  accomplish  the  same  purpose. 

We  have  seeii  that,  by  putting 

a;  =  |,    y  =  kx, 

tvhere  k  is  an  integer,  we  can  always  transform  the  given  equa- 
tion ivith  rational  coefficients  into  another  one  of  the  form 

(5)  ?/"  +  6t/"-i  +  c«/"-2  H \-ly  +  m  —  ^ 

which  has  the  folloiving  ttvo  properties  : 

(a")  its  coefficients  are  integers  ; 

(5)  the  coefficient  of  the  highest  power  of  the  unknown  quantity 
is  equal  to  unity. 

If  the  original  equation  has  a  rational  root,  (5)  must  also 
have  a  rational  root,  since  every  root  of  (5)  is  equal  to  k 
times  a  root  of  (1),  and  k  is  an  integer.  Conversely,  to 
every  rational  root  of  (5)  corresponds  a  rational  root  of  (1) 
by  means  of  (-3). 

Thus,  the  problem  of  finding  the  rational  roots  of  (1)  will  be 
solved  if  we  can  find  the  rational  roots  of  (5). 


Art.  110]      RATIONAL   ROOTS   OF   AN    EQUATION  179 

But  the  latter  may  be  found  with  comparative  ease,  on 
account  of  the  following  theorem. 

Jf  an  equation  has  the  properties  (a)  and  (5),  mentioned 
above,  any  rational  roots  which  it  may  have  must  be  integers. 

To  prove  this,  let  (5)  be  the  given  equation,  having  the 
properties  (a)  and  (/>).  If  this  equation  has  a  rational  root, 
let  us  denote  this  root  by 

(6)  y=^P 

9 

where  p  and  q  are  integers  without  a  common  divisor,  so 
that  the  fraction  p/q  is  in  its  lowest  terms.  If  (6)  is  a  root 
of  (5),  we  must  have 

9"  \    ^"~  9"  q         J 

or  E.  —  —  (bp''-'^  +  cp^-^i  +  •  •  •  +  Ipq""'^  +  mq"''^}. 

The  right  member  of  this  equation  is  an  integer.  The  left 
member  is  not  an  integer  unless  5-=  ±  1,  in  which  case  the 
root  (6)  is  itself  an  integer,  thus  proving  our  theorem. 

Thus,  it  only  remains  to  settle  the  question  whether  (5) 
has  any  integral  roots.  This  can  be  done  quite  easily  on 
account  of  the  following  theorem: 

Any  integral  root  of  an  equation,  of  the  form  (5),  tvith  inte- 
gral coefficients,  must  be  a  divisor  of  the  constant  term  m  of  the 
equation. 

In  fact  if  an  integer  y  satisfies  equation  (5),  we  have 

m  =  —  y  —  by'*-^  — ly  =  —  y{y''~^  -I-  ^i/"~^  H f-  0 

showing  that  w  is  a  product  of  two  integers  one  of  which  is 
equal  to  y. 

Thus,  if  we  wish  to  examine  the  given  equation  (1)  for 
rational  roots,  we  proceed  as  follows  : 

1.  If  the  given  equation  in  x  does  not  have  the  properties 
(a)  and  (5),  ive  transform  it  into  another  equation  in  y  which  has 


180  INTEGRAL   RATIONAL   FUNCTIONS         [Akt.  110 

these  properties  hy  putting  x  =  y/k^  using  the  smallest  value 
of  the  integer  k  which  ivill  accomplish  the  purpose. 

2.  The  resulting  equation  in  y  must  have  integral  roots  if 
it  has  any  rational  roots  at  all.  These  integral  roots  must  he 
divisors  of  its  constant  term.  Therefore,  we  test  each  of  the 
divisors  of  the  constant  term  of  the  equation,  to  see  ivhether  it 
is  or  is  not  a  root  of  the  equatiori  in  y. 

3.  From  every  integral  root  of  the  equation  in  y  which  is 
obtained  in  this  way,  we  find  a  rational  root  of  the  original 
equation  in  x  hy  dividing  it  hy  k. 

It  may  not  be  necessary  to  test  all  of  the  divisors  of  m  as  indicated 
in  No.  2.  If  some  of  them  lie  beyond  the  upper  limit  for  positive  roots, 
or  below  the  lower  limit  for  negative  roots  (see  Art.  105),  they  cannot  be 
roots  of  the  equation  and  need  not  be  tested.  Descartes's  rule  (Art.  106) 
may  also  frequently  be  used  to  reduce  the  number  of  trials.  Moreover, 
if  one  rational  root  has  been  found,  it  will  usually  be  advisable  to  make 
use  of  it  to  depress  the  given  equation  before  proceeding  farther. 

EXERCISE  XLVI 

Examine  the  following  equations  for  rational  roots. 
1.    108x3  -  ,54x2  +  45  a;  -  13  =  0. 

Solution.     We  first  write  this  equation  in  the  form 

(1)  x^-\x^+j\x-^^\  =  Q. 
If  we  put  X  =  y/k,  //  will  satisfy  the  equation 

The  smallest  value  of  k  which  will  make  the  coefficients  of  this  equation 
integers  is  A;  =  6.     Therefore  we  put  x  =  y/6  and  obtain  the  equation 

(2)  /  -  3  f  +  15  y  -  26  =  0 

for  y.  This  equation  has  properties  (a)  and  (b) ;  any  i"ational  root 
which  it  may  possess  must  therefore  be  an  integer,  and  moreover  a 
divisor  of  —  26.  The  only  integral  values  of  y  that  w^e  need  test  there- 
fore are  ±1,  ±2,  ±13,  ±  26,  since  these  are  the  only  integral  divisors 
of  -  26. 


Art.  Ill]     SUMMARY  OF  OPERATIONS   REQUIRED  181 

But  according  to  Descartes's  rule,  (2)  has  no  negative  roots.  It  suffices 
therefore  to  test  +  1,  +  2,  +  lo,  +  26.  We  begin  with  +  1.  Synthetic 
division  gives 

1      _  3     +1.5     -  26  |_1 

1  -    2     4-  13 
-  2     +13     -  13 

Therefore  +  1  is  not  a  rout  of  (2).     We  test  +  2  next. 

1      _  ;j      +15      -  2()  I  +2 

2  -    2     +26 


1-1+13  0 


Therefore  +  2  is  a  root  of  (2).     The  depressed  equation 

Z/-- 2/ +  13  =  0 

is  a  quadratic  whose  discriminant  is  equal  to  1  —  4  •  13  =  —  51,  which  is 
not  a  perfect  square.  Consequently  this  quadratic  has  no  rational  root, 
and  therefore  3/  =  +  2  is  the  only  rational  root  of  (2).  Since  x  —  y/Q, 
the  only  rational  root  of  (1)  is  x  —  1/3. 

2.  J-3  -  3  x'^  -  2  X  +  6  =  0.  6.  x<  -  6  x8  +  6  x2  +  5  X  +  12  =  0. 

3.  x3  -  8  x2  +  17  X  -  10  =  0.  7.  2  x-»  -  x3  -  5  x2  +  7  X  -  6  =  0. 

4.  x3  -  9  x2  +  23  X  -  15  =  0.  8.  x^  +  -i/-  -i'-  -  '/  ^  +  V  =  0. 

5.  3x3  +  8x2  + X  -  2  =  0.  9.  12x3  -  13x2  +  fx  -  ^  =  0. 

111.  Summary  of  the  operations  required  in  solving  an 
equation  with  given  numerical  coefficients.  The  application 
of  iSewton's  or  Horner's  method  is  likely  to  cause  trouble 
when  the  root  to  be  determined  is  a  multiple  root.  It  is 
advisable,  therefore,  to  apply  the  method  of  Art.  109  in  order 
to  detect  and  determine  any  possible  multiple  roots  before 
proceeding  farther.  It  will  then  be  easy  to  depress  the 
equation  (Art.  104).  dividing  by  (^x  —  ay  if  a  is  a  Ar-tuple 
root,  by  (x  —  by  if  b  is  an  Staple  root,  and  so  on.  The 
resulting  equation  will  have  only  simple  roots.  If  its  coeffi- 
cients are  rational  numbers,  it  should  be  tested  for  rational 
roots  (Art.  110). 

Descartes's  rule  (Art.  106)  may  then  be  used  to  find  an 
upper  bound  for  the  number  of  positive  and  negative  roots 


182  INTEGRAL   RATIONAL   FUNCTIONS         [Art.  112 

of  this  equation.  li  the  left  member  of  the  equation  be 
called  /(a;),  we  may  compute  the  value  of  f(x),  by  sjm- 
thetic  division  (iVrt.  93),  for  various  values  of  a;,  obtain- 
ing incidentally  an  upper  limit  for  positive  roots  (Art.  105), 
a  lower  limit  for  negative  roots  (Art.  105),  and  some  indica- 
tion as  to  the  location  of  the  roots.  If  we  wish,  these  calcu- 
lations may  be  used  for  plotting  the  graph  of  y  =f{x).  As 
soon  as  we  have  discovered  in  this  way  the  approximate 
location  of  a  root,  we  may  determine  its  position  as  accu- 
rately as  we  please  by  means  of  Newton's  or  Horner's  method 
(Arts.  100  and  101),  or  by  the  method  of  false  position  (Art. 
99). 

While  we  have  learned  in  Art.  109  how  to  avoid  the 
trouble  caused  by  multiple  roots,  there  remains  another  case 
which  requires  a  word  of  explanation.  If  two  roots  a  and  h 
are  very  close  together  without  coinciding  absolutely,  essen- 
tially the  same  difficulties  appear  in  applying  Newton's  or 
Horner's  method  as  in  the  case  of  a  double  root,  but  we  can- 
not use  the  same  method  for  overcoming  these  difficulties. 
The  methods  best  adapted  for  separating  the  roots  in  such 
cases  are  connected  with  a  theorem  of  Sturm's  and  cannot 
be  discussed  here.  See  Dickson's  Elementary  Theory  of 
Equations^  pag6  96. 

We  have  said  nothing  about  any  methods  for  calculating 
imaginary  roots.  Since  a  complex  quantity  contains  two 
real  numbers  (its  real  and  imaginary  components),  the  prob- 
lem of  calculating  a  complex  root  may  be  regarded  as  one 
involving  two  unknowns,  and  such  problems  are  reserved  for 
a  later  chapter. 

112.  Application  of  cubic  equations  to  floating  spheres.    All 

calculations  about  floating  bodies  are  based  upon  a  funda- 
mental law  usually  called  the  principle  of  Archimedes  after 
its  discoverer.  Archimedes  (287-212  b.c),  who  is  gen- 
erally regarded  as  the  greatest  mathematician  of  antiquity, 
lived  in  Syracuse,  which  was  then  a  prosperous  city  of  Sicily 
inhabited  by  colonists  who  had  come  from  Greece.  Accord- 
ing to  a  familiar  story,  Hiero,  king  of  Syracuse,  had  given 


Art.  n-2]      APPLICATION  OF   CUBIC   EQUATIONS  183 

orders  to  a  goldsmith  to  make  a  crown  for  him,  and  had 
given  him  the  necessary  amount  of  gold  carefully  weighed. 
When  the  crown  was  delivered,  its  weight  was  found  to  be 
correct,  but  the  suspicion  arose  that  the  goldsmith  had  de- 
frauded the  king  by  replacing  some  of  the  gold  by  an  equal 
weight  of  silver.  But  how  was  this  suspicion  to  be  verified? 
Knowing  the  great  reputation  of  Archimedes,  the  king  laid 
the  case  before  him,  and  Archimedes  promised  to  make  an 
attempt  to  solve  the  problem.  A  short  time  afterward,  while 
in  the  public  baths  of  Syracuse,  he  observed  that  the  water 
seemed  to  exert  an  upward  pressure  upon  his  body,  and  that 
this  pressure  increased  or  decreased  according  as  more  or  less 
of  his  body  was  immersed.  Recognizing  the  bearing  of  this 
observation  on  the  problem  of  Hiero's  crown,  he  rushed 
out  into  the  street  shouting,  "  I  have  found  it,  I  have  found 
it." 

His  solution  of  the  problem  was  as  follows.  He  weighed 
out  a  quantity  of  gold  and  an  equal  weight  of  silver,  the 
weighing  being  performed  in  air  as  usual.  He  then  attached 
these  equal  weights  of  silver  and  gold  to  the  two  ends  of  a 
bar  with  equal  arms,  which  would  therefore  be  in  complete 
equilibrium.  He  then  placed  a  vessel  filled  with  water 
underneath  the  bar,  so  that  both  the  gold  and  the  silver  were 
covered  with  water.  The  silver  now  seemed  to  weigh  less 
than  the  gold.  This  being  established,  the  problem  of  the 
crown  could  be  solved  easily.  If  on  a  balance  the  crown  was 
measured  against  an  equal  weight  of  gold,  and  then  the 
whole  was  immersed  in  water,  the  gold  would  outweigh  the 
crown  if  the  goldsmith  had  been  dishonest. 

The  fundamental  principle  of  Hydrostatics  discovered  by 
Archimedes  may  be  stated  as  follows.  When  a  body  is 
immersed  in  water,  the  water  exerts  an  upward  pressure 
upon  it  which,  either  partially  or  completely,  counteracts  the 
downward  tendency  due  to  gravity. 

Consequently  a  body  will  loeigh  less  when  immersed  in  water 
than  in  air.  This  loss  in  weight  is  exactly  equal  to  the  weight 
of  the  water  which  the  body  displaces. 


184 


INTEGRAL   RATIONAL   FUNCTIONS         [Art.  112 


In  the  case  of  a  floating  boch%  the  upward  pressure  of 
the  water  just  balances  the  downward  effect  of  gravity. 
Therefore  : 

The  total  iveight  of  a  float iny  body  is  ejjual  to  the  weic/ht  of 
the  water  which  it  displaces. 

Since  the  weight  of  a  body  varies  as  its  mass  (see  Art.  80) 
we  may,  if  we  prefer,  also  state  this  law  as  follows. 

TJie  total  mass  of  a  floatijig  body  is  equal  to  the  mass  of  the 
water  ivhich  it  displaces. 

We  proceed  to  make  an  application  of  this  principle.     Fig- 
ure 52  represents  the  cross  section  of  a  sphere  of  radius  r  float- 
ing in  water  and  immersed  to  a  depth 
AB  =  h.     The  volume  of  the  sphere  is 

The  volume  of  the  submerged  por- 
tion of  the  sphere  is  of  course  equal  to 
the  volume  of  the  water  which  has  been 
displaced  by  tlie  floating  body.     If  we 


FiU.  o2 


call  this  volume  -y',  we  shall  have 

(2)  v'  =7rA2(>- 1  /O-t 

If  we  use  the  centimeter  as  unit  of  length  and  the  gram  as 
unit  of  mass,  the  density  of  water  is  equal  to  unity  (see  Art. 
44),  so  that  the  mass  (in  grams)  of  the  water  displaced  will 
be 

(3)  m'  =  'Kli\r  -^h). 

If  the  sphere  is  composed  of  material  of  density  p,  the  mass 
of  the  spliere  will  be 

(4)  w  =  I  'TTT^P- 

*  This  is  the  formula  for  the  volume  of  a  sphere  of  radius  /■. 
t  This  is  the  formula  for  the  volume  of  a  spherical  segment   in  terms  of  the 
altitude  h  and  the  radius  r  of  the  sphere. 


AiiT.  113]       APPLICATION   OF   CUBIC    EQUATIONS  185 

According  to  the  principle  of  Archimedes,  w'  must  be  equal 
to  m  if  tlie  sphere  floats,  that  is, 

'rrh^(r  —  J  A)  =  |  Trr^p, 

whence  ^^(3  r  —  k)  =  A  r^p, 

or 

(5)  A3  _  3  rJfi  +  4  rV  =  0. 

Since  the  density  p  of  u  substance,  when  measured  in 
terms  of  grams  per  cubic  centimeter,  is  tlie  same  number  as 
the  specific  gravity  of  that  substance  (Art.  44),  we  may  state 
our  result  as  follows: 

If  a  solid  sphere  of  radius  r,  lohich  is  composed  of  material 
whose  spenfic  gravity  is  equal  to  p,  floats  upon  water,  the  depth 
h  to  which  it  will  sink  into  the  water  is  a  root  of  the  cubic 
equation  (-') ). 

Ill  ;>|i|ilyini:;-  (.')).  it  is  not  necessary  to  express  )•  ami  li  in  centimeters; 
tlit'V  may  lie  expressed  in  any  conveiiieut  unit,  but  of  course  the  same 
unit  must  he  used  for  both.  Tlie  reason  for  this  is  apparent  from  the 
equation  (5)  itself.  Every  term  of  (.5)  has  the  dimension  L^  since  p,  the 
specific  gravity,  is  an  abstract  number. 

113.   Application    of     cubic     equations     in    Trigonometry. 

From  the  addition  formulas 

^  sin  (a  +  /3)  =  sin  a  cos  (3  -f-  cos  a  sin  yS, 

cos  (a  +  /S)  =  cos  «  cos  /3  —  sin  «  sin  /3, 

we  obtain  the  familiar  formulas 

(2)  sin  2^  =  2  sin  6  cos  d,     cos  2  (9  =  os^  0  -  sin2  0 

by  putting  u  =  /3  =  6.     If  now  we  put  in  (1),  a=  '2  6.  /3  =  d 
and  make  use  of  (2),  we  easily  find 

sin  :5  6  ^  :]  s'\n  0  v.os^  6  -  sin3  6, 
cos  3  ^  =  cos3  ^  —  3  sin^  6  cos  6. 
Since  we  have  sin^^  +  cos^^  =  1, 

we  may  write  instead  of  (3), 

,  sin  3  ^  =  3  sin  ^  -  4  sin^  9, 

^  ^  cos  3  ^  =  4  cos3  ^  -  3  cos  6. 


186  INTEGRAL   RATIONAL   FUNCTIONS         [Art.  113 

If  in  these  equations  we  regard  sin  30  and  cos  3  6  as  known.,  the 
values  of  sin  6  aiid  cos  6  may  each  be  obtained  by  solving  a  cubic 
equation. 

This  fact  has  an  important  bearing  on  the  idiiwous  problem  of  the  trisec- 
tion  of  an  angle.  It  can  be  shown  that  the  solution  of  this  problem  is 
equivalent  to  that  of  finding  a  construction  for  the  sine  of  one  third  of 
the  given  angle.  It  can  further  be  shown  that  the  cubic  equation  (4) 
cannot  be  solved  by  expressions  involving  only  square  roots.  Finally  it 
can  be  shown  that  a  construction  in  the  sense  of  elementary  geometry, 
which  only  makes  use  of  the  ruler  and  compasses  as  instruments,  can  only 
be  performed  when  the  corresponding  problem  formulated  algebraically 
is  solvable  by  expressions  involving  only  square  roots.  Consequently  the 
problem  of  trisecting  an  angle  of  any  size  bij  a  ruler  and  compass  construction 
is  insolvable.  This  does  not  mean  that  an  angle  can  not  be  trisected.  It 
merely  means  that  it  cannot  be  trisected  with  the  help  of  ruler  and  com- 
passes alone. 

The  same  remark  applies  to  the  problem  of  the  dup>Ucation  of  the  cube, 
the  so-called  Delian  problem.* 

EXERCISE  XLVII 

1.  How  deep  will  a  sphere  of  yellow  pine  one  foot  in  diameter  sink  in 
water,  if  the  specific  gravity  of  yellow  pine  is  0.6.57?  Compute  to  three 
significant  figures. 

2.  How  far  will  a  cork  sphere  two  feet  in  diameter  sink  in  water,  if 
the  specific  gravity  of  cork  is  0.24?     Compute  to  two  significant  figures. 

3.  Apply  the  principle  of  Archimedes  to  a  floating  cube,  taking  for 
granted  that  it  will  float  with  its  upper  face  in  a  horizontal  position. 
Find  a  formula  for  the  depth  to  which  it  will  sink. 

4.  Apply  the  principle  of  Archimedes  to  a  rectangular  parallelopiped 
floating  in  a  horizontal  position.  Find  a  formula  for  the  depth  to  which 
it  will  sink. 

5.  Show  how  to  modify  formula  (.5)  of  Art.  112  if  the  sphere  floats 
upon  some  fluid,  not  water,  of  specific  gravity  p'.  Apply  your  result  to 
a  .sphere  of  iron  of  radius  6  inches,  floating  on  mercury.  The  specific 
density  of  iron  and  of  mercury  are  7.2  and  13.6  respectively. 

6.  Given  sin  30°  =  \.  Make  use  of  Art.  113  to  calculate  the  sine  of 
lO""  to  three  decimal  places.  Compare  your  result  with  that  given  in  a 
table  of  natural  sines. 

*  For  a  detailed  discussion  of  these  and  related  questions  consult  Klein's 
Famous  Problems  of  Elementary  Geometry,  translated  by  Beman  and  Smith,  or 
the  article  by  Dickson  on  Const nirtloits  vlth  Ruler  and  Compasses  in  the  Mono- 
graphs on  Topics  of  Modern  Mathematics  edited  by  J.  W.  A.  Young. 


CHAPTER   V 

INTEGRAL  RATIONAL  FUNCTIONS  OF  THE  nth  ORDER. 
THE  PROBLEM  OF  THE  ALGEBRAIC  DETERMINATION 
OF   THEIR  ZEROS.    AND    THEIR  GENERAL   PROPERTIES 

114.  Distinction  between  the  algebraic  and  numerical  solu- 
tion of  an  equation.  We  have  shown  how  the  real  roots  of  an 
equation  of  the  form 

(1)  Ax" +  Bx''-'^  + '■■  +Lx  + M=0 

may  be  determined,  if  the  coefficients  A,  B,  •'  L,  Mare  given 
numbers.  But  our  solution  of  the  problem  was  arithmetical 
rather  than  algebraic,  since  we  did  not  find  a  general  formula 
for  the  value  of  the  roots  of  (1)  in  terms  of  the  coefficients. 
In  two  cases,  however,  namely,  when  the  given  equation  is  of 
the  first  or  second  degree,  we  did  find  such  general  formulas. 
Thus  we  have  actually  accomplished  the  algebraic  solution 
of  equations  of  the  first  and  second  degree.  (See  Chapters 
II  and  III.)  We  shall  now  show  that  there  are  some  other 
cases  in  which  we  are  able  to  solve  an  equation  algebraically . 

115.  The  equation  a:«-1  =  0.  If  ^  =  1,  B=C=  ••  =X  =  0, 
iHf  =  —  1,  equation  (1)  of  Art.  114  reduces  to 

(1)  a;"  -  1  =  0  or  x"  =  1. 

If  n  is  an  odd  integer,  the  only  real  number  which  satisfies 
this  equation  is  a;=  1.  If  n  is  even,  the  equation  has  two 
real  roots,  namel}^  x  =  +  1  and  x  =  —\. 

The  question  now  arises  whether  (1)  may  not  also  have 
some  complex  roots.  This  question  may  also  be  formulated 
thus.  Can  the  wth  power  of  a  complex  number  be  equal  to 
unity  ? 

187 


188  INTEGRAL   RATIONAL   FUNCTIONS         [Art.  11(3 

116.  The  TJth  power  of  a  complex  number.  In  order  to 
settle  this  question,  we  iimst  first  learn  how  to  determine  the 
nth  power  of  a  given  complex  number.  The  easiest  way  to 
accomplish  this  is  to  make  use  of  the  geometric  representa- 
tion of  a  complex  number  as  a  vector,  which  was  explained 
in  Art.  24,  and  the  definition  of  multiplication  of  two  com- 
])lex  numbers  as  given  in  Theorem  I,  of  Art.  31.  Let  r^  and 
r^  be  the  moduli  of  two  complex  numbers,  and  let  6^  and  6^ 
he  their  amplitudes.  Then,  according  to  the  theorem  just 
quoted,  the  product  will  be  a  complex  quantity  whose  modu- 
lus is  equal  to  r^r^  and  whose  amplitude  is  equal  to  ^1  +  ^2- 

If  rj  and  r^  are  both  equal  to  r,  and  6-^  and  6^  are  both  equal 
to  6,  the  product  will  be  the  square  of  that  complex  number 
which  has  r  as  its  modulus  and  6  as  its  amplitude.  There- 
fore the  square  of  this  complex  number  will  have  r  •  r  =  r^ 
for  its  modulus  and  0-^6=20  as  its  amplitude.  In  the 
same  way  w^e  see  that  the  cube  of  the  given  complex  number 
will  have  r^  for  its  modulus  and  3  6  for  its  amplitude.  In 
general,  we  obtain  the  following  result. 

Jfr  is  the  modulus  and  6  the  amplitude  of  the  complex  num- 
ber a  +  bi,  the  nth  power  of  a  +  bi  will  have  r"  for  its  modulus 
and  nd  for  its  amplitude. 

The  same  result  may  be  obtained  by  using  the  polar  form  of  the  com- 
plex number  (Art.  30).     If  we  write 

(1)  X  =  a  +  hi  =  r(cos  0  +  i  sin  6), 

we  find  first 

x^  =  X-  x  =  r-  r[cos(^  +  6)+  /sin(^  +  6)},  (Equation  (:3),  Art.  31) 
or 

(2)  x2  =  r2  (cos  20  +  1  sin  2  6) . 

Again  we  have 

x^^  x-^x=  r-  ■  r  [cos(L>  $  +  0)  +  i  sin(2  6  +  6)], 
or 

(3)  .r8  =  r3(cos  :i6  +  i  sin  3  6)  ; 

and  continuing  in  this  way,  we  finally  obtain  the  formula 

(4)  x"  =  r"(cosn^  +  i  sin  nd). 


1.   r=\,  6=  9fr. 

5.  r=l,  6=  120°. 

2.    ;•  ^  ^.  6  =  90^ 

6.  r  =  I,  ^  =  120\ 

3.    r  =  i,  ^  =  90  '. 

7.  r  =  1,  ^  =  240'^ 

Aim.  117]  THE    COMPLEX    ROOTS   OF    UNITY  189 

Coiiip.irisoii  <>t'(l)  ami  (1)  gives  ri.se  to  an  iiiiiiortant  re.sult.  If,  in 
(4),  we  .sul»stitiit«'  for  x  its  value  from  (1).  and  then  divide  l)Oth  meniber.s 
of  the  resulting-  equation  l>y  '",  we  tind 

(5)  (cos  6  +  ;  sin  6)"  —  cos  nd  +  /  sin  nd, 

a  remarkable  equation  usually  known  us  De  Moivre's  formula. 

EXERCISE    XLVIII 

Plot  the  vectors  wliich  corre.spond  to  the  data  given  in  the  following 
examples  and  then  plot  their  squares,  cubes,  and  fourth  power.s. 

9.  r=\,  e  =  240°. 

10.  r  =  \,  6  =  60^ 

11.  r  =  1,  ^  =  70°. 

Q«no 
4.    r=\,  6=  120-.  Q.  r=l,  e  =  2i(P.  12.  r=l,  6  =  '^". 

117.  The  complex  roots  of  unity.  We  are  now  prepared  to 
answer  the  question  raised  in  Art.  115,  whether  the  wth  root 
of  a  complex  number  may  be  equal  to  unity.  In  Fig.  53,  the 
vector  (9^j,one  unit  long  in  the  direction  of  the  positive  aj-axis, 
represents  tlie  complex  number  1=.  1  +  0  ■  i.  Let  us  draw  a 
circle  of  unit  radius  with  0  as  center,  so  that  A-^  will  be  upon 
its  circumference.  Let  us  divide  the  circumference  into  n 
equal  part.s,  Aj^  being  one  of  the  points  of  division,  and  the 
others  being  denoted  by  A^,  A^,  •••  -4„.  (Li  Fig.  53,  we  have 
chosen  n  =  8.)  Then,  each  of  the  n  vectors  OA^  OA^.  ^-^z'  "' 
OAn  represents  a  complex  number  ivhose  nth 
power  is  equal  to  unity. 

PiiOfJF.  Consider  OA^  It  represents 
the  complex  (piantity  who.se  modulus  is  1 
and  whose  amplitude  is  0°.  According 
to  Art.  31,  the  nth  power  of  this  complex 
quantity  Mill  have  as  its  modulus  1"  =  1 
and  its  amplitude  w  x  0°  =  0°.  Therefore 
the  nth  power  of  this  complex  quantity  is  0(|ual  to  unity,  as 
is  also  evident  by  calculation. 

Consider  OA,^.,  which  represents  a  complex  quantity  whose 
modulus  is  1,  and  whose  amplitude  is  360'^//«.  (In  the  figure 
this  angle  is  equal  to  45°.)     According  to  Art.  31,  the  nth 


190  INTEGRAL   RATIONAL   FUNCTIONS         [Art.  117 

power  of  this  complex  quantity  will  have  as  its  modulus 
1"  =  1,  and  the  value  of  its  amplitude  will  be  w(360°/w)  =  360°. 
Therefore  the  nth  power  of  the  complex  quantity  represented 
by  OA2  is  the  complex  quantity  represented  by  OA^  which 
is  1  +  0  .^■  =  1. 

In  the  same  way  we  can  show  that  each  of  the  n  vectors 
mentioned  represents  a  complex  quantity  whose  nth  power  is 
equal  to  unity. 

It  is  easy  to  shotv  further  that  the  n  complex  quantities  repre- 
sented by  OA-^,  OA^,  •  •  •  OAn  are  the  only  ones  whose  nth  powers 
are  equal  to  unity. 

For,  if  r  is  the  modulus  and  6  the  amplitude  of  a  complex 
quantity  whose  nth  power  is  equal  to  1  +  0  •  z,  r"  must  be  equal 
to  the  modulus  of  1  +  0  •  z,  and  n9  may  differ  from  the  ampli- 
tude of  1  +  0  ■  i  only  by  an  integral  multiple  of  360°. 
(See  Art.  25,  last  statement.)     Consequently  we  must  have 

(1)  r"  =  1,     nO  =  A"  •  360°, 

where  k  may  be  zero  or  any  positive  or  negative  integer. 
Now  r  is  the  modulus  of  a  complex  number  and  is  therefore 
positive,  by  definition.  (See  Art.  23.)  Since  r  is  positive, 
and  since  the  onl}'  positive  number,  whose  nth  power  is  equal 
to  unity,  is  unity  itself,  we  find  from  (1) 

(2)  r=l,     ^  =  ^^^, /t  =  0,  ±1,  ±2,  ±3,  .... 

n 

If  in  these  equations  we  put  in  succession  ^  =  0,  1,  2, 
•  "  n  —  1,  we  obtain  precisely  the  n  vectors  OA^,  OA^,  •••  OA^ 
of  Fig.  53.  If  we  give  any  other  integral  value  to  k,  for 
instance  a  positive  value  greater  than  n  —  1  or  a  negative 
value,  we  only  obtain  one  of  these  same  vectors  over 
again. 

A  number,  real  or  complex,  whose  nth  power  is  equal  to 
unity  is  called  an  nth  root  of  unity.  We  have  shown  that 
there  exist  exactly  n  distinct  nth  roots  of  unity.  Tliey  are  the 
n  complex  'numbers  which  are  represented  by  the  n  vectors  OA^, 


Art.  118]  THE   COMPLEX   ROOTS   OF   UNITY 


191 


OA^^  •••  OAn  of  Fig.  53.      Therefore  the  modulus  of  every  nth 
root  of  unity  is  equal  to  1.      Their  amplitudes  are 

/Qx       no  360°  .-,360°   ..360°         ,360°         .        ...360° 

(3)       0°, ,2 ,3 ,---k ,  .-(m-I) 

n  n  n  n  n 

respectively. 

We  may  write  out  the  value  of  each  of  these  nth  roots  of  unity  in  its 
polar  form  at  once.  A  complex  number  of  modulus  /•  and  amplitude  $ 
may  be  written  in  the  form 

r(cos  0  +  i  .sin  6).  (See  Art.  30) 

Consequently  the  n  roots  of  unity,  whose  amplitudes  are  the  angles  listed 
in  (3),  may  be  expressed  as  follows  : 

.r^  =  cos  0'^  +  i  sin  O^  =  1, 

360^  ,   .  .    360° 
Xj  =  cos 1-  ism , 


(4) 


Xk  —  COS  k )  +  I  sm 


.n  \k—). 


X„-l 


(«  -  1)360°  ,    .    .     (n-  1)360° 

COS  ^^ ^ h  I  sin  '^ '- • 


118.  Numerical  expressions  for  the  complex  roots  of  unity 
for  n  =  2,  3,  4.  It  is  evident  that  the  two  square  roots  of 
unity  are  +  1  and  —  1.  In  order  to  obtain  expressions  for 
the  cube  roots  of  unity,  consider  Fig.  54, 
where  the  vectors  OA^,  OA^,  and  OA^ 
represent  these  cube  roots.  Since  the 
angle  between  OA^  and  0^12  is  120°,  OB 
the  bisector  of  this  angle  will  make  an 
angle  of  60°  with  OAo  and  the  triangle 
OA^B,  isosceles  by  construction,  will  have 
to  be  equiangular  and  therefore  equilat- 
eral. Moreover,  each  of  its  sides  is  of  unit  length.  Con- 
sequently the  length  of  A^C  is  equal  to  -|,  and  that  of  OC  is 
equal  to  Vl  -QY  =  V|  =  1 V3.  Since  A^  is  to  the  left  of 
the  y-axis,  its  abscissa  is  negative.  Therefore  the  coordi- 
nates of  A^  are  a:  =  -  .},  y  =  -h  ^V3.      If  a  vector  is  in  its 


Fig.  54 


f 

192  INTEGRAL   RATIONAL   FUNCTIONS         [Art.  118 

standard  position  (see  Art.  2-3)  and  the  coordinates  of 
its  terminus  are  (a:,  ?/),  the  vector  represents  the  complex 
number  x-{-yi.  Consequently  the  complex  number  repre- 
sented by  OA^  is  i       i  •   /- 

Similarly  we  find  that  OA^  represents  the  complex  number 

-I- I  iV8. 
Therefore,  the  three  cube  roots  of  unity  are 

(1)  1'  -  i  +  i  ^V8,    -l-\  ^  V3. 

A  figure  may  be  constructed  very  easily  to  represent  the 
four  fourth  roots  of  unity.  Inspection  of  such  a  figure  shows 
that  the  four  fourth  roots  of  unity  are 

(2)  1,  i  -  1,  -  L 

These  same  results  may  also  be  obtained  without  any  use 
of  geometr}'.  The  cube  roots  of  unity  are  those  numbers 
which  satisfy  the  equation 

(3)  x^=l  or  x^  -1  =  0. 

This  equation  obviously  has  the  root  x  =  1.  Therefore 
.c  —  1  is  a  factor  of  x^  —  1  and,  in  fact,  we  find 

.i-3_  1  =(.f-  l}{x^+  J+1). 

Thus  the  other  roots  of  the  equation  (3),  that  is,  those 
which  are  different  from  1,  must  satisfy  the  quadratic  equa- 
tion „  -, 

If  we  solve  this  quadratic,  we  find  precisely  the  second  and 
third  of  the  expressions  (1). 

Similarly  the  fourth  roots  of  unity  are  the  roots  of  the 
eti  nation 

x^  =  1  or  2-4  —  1  =  0, 

which  may  be  factored  into 

(x^  -  1  )(.r2  +l^==(x-l)(x+  l)(x  +  0(.r  -  i)  =  0, 

showing  that  its  four  roots  are  precisely  the  four  complex 
numbers  (2). 


Art.  119]       CONSTRUCTION   OF   REGULAR   POLYGONS       193 

We  shall  frequently  denote  the  second  of  the  three  cube  roots  of 
unity^  namely^  —  \  +  \  iV8,  by  the  letter  (o.  It  follows  either 
without  calculation  from  geometry,  or  by  direct  multiplica- 
tion, that  the  third  cube  root  of  unity  will  then  be  equal  to  tu'-^. 

Thus  we  liave 

(4)  w  =  -  .}  -f  .]  iV-),  ^-  =  -  2  -  2  ''^•^'  <«'^  =  "1- 

We  may  find  a  numerical  expression,  at  least  approxi- 
mately, for  any  wth  root  of  unity  by  drawing  a  figure,  such 
as  Fig.  53,  accurately  to  scale  and  then  measuring  the  co- 
ordinates of  the  points  A^,  A^^  •••  -4„.  Or  else  we  may  use 
the  trigonometric  expressions  (4)  of  Art.  117,  making  use 
of  a  table  of  natural  sines  and  cosines  for  the    purpose  of 

1     ^.         .     360°  360°      , 

evaluating  sin ,  cos  ,  etc. 

n  n 

EXERCISE  XLIX 

1.  Draw  a  figure  representing  the  five  fifth  roots  of  unity  and  write 
each  of  them  in  the  form  x  +  yi,  the  vahies  of  x  and  //  heing  obtained  to 
two  decimal  places  by  measurement. 

2.  Do  the  same  thing  for  the  seven  seventh  roots  of  unity. 

3.  Find  the  exact  expressions  for  the  six  sixth  roots  and  the  eight 
eighth  roots  of  unity. 

4.  Prove  tliat  each  of  the  imaginary  cnbe  roots  of  unity  is  the  square 
of  the  other. 

5.  Let  Xj  be  that  nt\\  root  of  unity  wliose  amplitude  is  equal  to 
:}60Y«,  and  let  Xf.  be  that  ?jth  root  of  unity  whose  amplitude  is  equal  to 
k  -  :}60"/;*.     Prove  that  x^.  =  xi*'. 

6.  Review  the  construction  of  a  regular  pentagon  from  ])lane  geom- 
etry. By  translating  the  steps  of  that  construction  into  algebra  show 
that  one  of  the  fifth  roots  of  unity  is  equal  to 


j(  V5  -  1)  +  -  VlO  +  2x^5  . 
4- 

119.  Construction  of  regular  polygons.  Our  discussion  of  the  equation 
.,"  -  1  —  0  suffices  to  show  how  very  closely  its  solution  is  connected 
with  the  problem  of  dividing  the  circumference  of  a  circle  into  n  equal 
parts  or,  what  amounts  to  the  same  thing,  with  the  problem  of  construct- 
ing a  regular  «-gon.  On  account  of  this  connection  the  equations  of  the 
form  x"  —  1  =  0  are  frequently  called  cyclotomic  equations. 


194  INTEGRAL   RATIOXAL   FUNCTIONS         [Akt.  120 

It  is  a  familiar  fact  of  elementary  geometry  that  it  is  possible  to  con- 
struct a  regular  polygon  of  71  sides  with  the  help  of  ruler  and  compasses 
if  M  is  a  power  of  2,  if  n  is  equal  to  3  or  5,  or  if  n  is  a  product  of  any 
two  or  three  numbers  of  this  kind.  All  of  this  was  known  to  the 
ancients  and  was  recorded  by  Euclid.  No  further  progress  in  this 
direction  was  made  for  two  thousand  years,  until  Gauss  in  the  early 
part  of  the  nineteenth  century  proved  that  regular  polygons  of  17,  257, 
or  65,537  sides  may  also  be  constructed  with  ruler  and  compasses. 
More  specifically  the  substance  of  Gauss's  theorem  is  as  follows  :  If  n  is 
a  number  which  can  be  expressed  as  a  product  of  any  power  of  2  and  one  or 
several  other  distinct  factors,  each  (f  which  is  a  prime  number  of  the  form 

(1)  22*  +  1, 

then  and  only  then  will  it  he  possible  to  construct  a  regular  polygon  of  n  sides 
loith  the  help  of  ruler  and  compasses. 

The  numbers  of  the  form  (1)  which  are  obtained  by  putting  h  =  0, 
1,  2,  3,  4  are  in  order  3,  5,  17,  257,  65,537  and  all  of  these  are  actually 
prime  numbers.  The  numbers  (1)  which  correspond  to  h  =  5,  6,  7,  8,  9 
are  known  not  to  be  primes. 

For  a  proof  of  the  above  theorem  consult  the  articles  quoted  in  the 
footnote  on  page  186. 

120.  The  equation  jr"  —  a  =  0.  If  a  is  a  positive  number, 
this  equation  has  one  positive  root  which  is  usually  denoted  by 

(1)  Xq  =  </a. 

Let  x-^  be  any  other  solution  of  the  equation.     Then  we  shall 

have 

a^i"  =  a,  Xq^  =  a, 

whence  by  division 

^  =  ^orf^]   =1. 


^0       ^        ^^0 

Therefore,  if  we  denote  the  ratio  of  x^/xq  by  y,  we  see  that  y 
must  be  a  root  of  the  equation 

(2)  r  =  1 

and  that  x^  =  X(^. 

But  we  have  just  learned  that  (2)  has  n  distinct  solutions, 
only  one  of  which  is  equal  to  unity.     (See  Art.  117.) 

Thus,  if  a  is  a  positive  number,  the  equation  x"  —  a=  0  has 
one  positive  solution,  Xq=  -y/a,  and  n—  1  other  solutions,  each  of 


Art.  120]  THE   EQUATION   r«  -  a  =  0  195 

which  may  he  obtained  from  the  positive  solution  by  multiplica- 
tion with  one  of  the  nth  roots  of  unity. 

Thus  the  equation  x"  —  a  =  0  has  n  roots.  If  a  is  positive 
and  if  n  is  odd,  only  one  of  these  roots  is  real  ;  if  w  is  even, 
two  of  the  n  roots  are  real,  one  being  positive  and  one  nega- 
tive. The  moduli  of  all  of  the  roots  are  equal  to  each  other 
(each  equal  to  -y/a)  ;  only  their  amplitudes  differ.  If  we 
represent  the  w  roots  of  this  equation  geometrically  as  in 
Art.  117,  we  find  exactly  the  same  results  as  for  the  equation 
a:"  —  1  =  0,  with  this  one  exception  ;  the  end-points  A^,  A,^^ 
•  ••  An  of  the  n  vectors  which  represent  the  n  roots  are  on 
the  circumference  of  a  circle  of  radius  ^a  instead  of  being 
on  a  circle  of  radius  unity. 

If  the  quantity  a  is  not  positive,  nothing  very  essential  is 
changed  in  our  argument,  except  that  in  this  case  there  will 
usually  not  be  even  a  single  real  root.  To  be  quite  general, 
let  a  be  any  complex  number,  and  let  p  be  its  modulus  and 
a  its  amplitude.  On  the  other  hand,  let  r  be  the  modulus 
and  6  the  amplitude  of  the  unknown  complex  number  a-,  for 
which 

(3)  a;"  =  a. 

The  modulus  and  amplitude  of  a:"  will  be  r"  and  n6  respec- 
tively. (See  Art.  116.)  Consequently,  according  to  the 
last  statement  of  Art.  25,  the  equation  (3)  will  be  possible 
only  if 

r"(the  modulus  of  a;")  is  equal  to  p  (tlie  modulus  of  a) 
nd  (the  amplitude  of  a:")  differs  from  a  (the  amplitude  of  a) 
at  most  by  an  integral  multiple  of  360°  ;  that  is,  if  and  only 

if 

r-  =  p,  ne  =  a  +  k-  360°,  /r  =  0,  ±1,  ±2,  ±  3,  ...  , 

whence  follows 

r  =  ^/p,  ^  =  «-t-yfc'^— ,  /t  =  0,  ±1,  ±2,..., 
n  n 

since  r  and  p  are  both  positive  numbers  (see  definition  of 
modulus  of  a  complex  number  in  Art.  23).     As  in  Art.  117 


196  INTEGRAL   RATIONAL   FUNCTIONS         [Art.  120 

all  of  the  distinct  values  of  6  are  obtained  if  we  allow  k  to 
assume  the  values  0,  1,  2,  •••  n—  1. 

We  have  the  following  final  result.  If  a  is  a  complex 
number  whose  modulus  is  equal  to  p  and  whose  amplitude  is 
equal  to  a,  the  equation  a:"  —  a  =  0  has  n  distinct  roots.  All  of 
these  roots  have  the  same  modulus.,  namely  V/o,  and  the  ampli- 
tudes of  the  roots  are 

...         a    a  ,  360°     «  ,    ,^360°  «  ,    .         ...300° 

(4)  ,  ~H ,-+2 ,...,-+(^1-1)- — - 

n    n         n       n  n  n  n 

respectively. 

If  we  plot  these  n  roots  as  vectors  in  their  standard  posi- 
tion, the  termini  of  these  vectors  will  again  be  the  vertices 
of  a  regular  n-gon,  inscribed  in  a  circle  of  radius  V/a  with 
the  origin  as  center.  But  in  general  this  inscribed  polygon 
will  have  none  of  its  vertices  on  the  a:- axis,  as  is  shown  by 
the  values  (4)  of  the  amplitudes  of  the  7i  vectors. 

The  n  roots  of  equation  (■))  may  be  written  as  follows,  if  we  make 
use  of  the  polar  form  of  a  complex  number. 

«/  r      «  ,  .  .    «n 

Xq  =  vpl  cos  — I- 1  sm  -    , 

(,-})    <j     xjt  =  Vp    cos  I  -  +  A; j  +  I  sm  (  -  +  k 1    , 

x„-,  =  Vp    cos  I  -  +  (n  -  1) +  <  sin    -  +  (n  -  1) . 

L        \n  n    I  \n  n    IJ 

EXERCISE   L 

Find  the  modulus  and  argument  of  each  of  the  roots  of  each  of  the 
following  equations.  Represent  the  corresponding  complex  numbers 
graphically  as  vectors.  Measure  the  x  component  and  //  component  of 
each  of  these  vectors,  and  use  the  result  to  find  an  approximate  expres- 
sion of  the  form  x  +  yi  for  the  corresponding  complex  number.  Also 
find  exact  expressions  of  this  form  for  these  complex  numbers,  except  in 
the  case  of  Examples  9  and  10.  Students  who  have  studied  trigonometry 
should  also  express  each  of  these  complex  numbevs  in  its  polar  form. 

1.  X*  -  Ki  =  0.  3.    x^  +  16  =  0.  5.    a;3  _  8  =  0. 

2.  X*  -  81  =  0.  4.    X*  +  81  =  0.  6.   x^  -  27  =  0. 


Am.  121]  TIIK   CUBIC   EQUATION  197 

7.  .,-3  +  8  =  0.  9.    x-5  -  32  =  0.  11.    x2  -  /  =  0. 

8.  x^  +  27  =  0.  10.    j'"  +  32  =  0.  12.    x^  +  i  =  0. 

121.   The  cubic  equation.     The  general  cubic  equation 

(1)  Ar"^  +  Bx^  +  Cx+  I)=Q 

may  always  be  reduced  to  the  form 

( -1)  j^  +  ha^  +  ex  +  d  =  0 

by  dividing  by  A.  But  equation  (2)  may  be  reduced 
further  by  a  simple  transformation.     Let  us  put  in  (2) 

X  ^=  y  +  k  or  y  =  x  —  k, 

where  k,  as  yet  uid-cnown,  is  to  be  chosen  in  such  a  way  as  to 
simplify  the  resulting  cubic  equation  in  y.      We  have 

x=  y  +  k^ 
a^=  y^  +  2  ky  +  k\ 
3^  =  y-i  +  8  ky'^  +  :'>  ^V  +  ^3. 

If  we  substitute  these  values  of  x^,  x\  and  x  in  (2),  and  ar- 
range the  result  according  to  descending  powers  of  y.  we  find 

(3)  y^  +  (Bk  +  b)y'-+Qlk^+-2kb  +  c-)y  +  k^  +  hk^  +  ek  +  d  =  0. 

We  may  now  choose  k  in  such  a  way  as  to  make  the  coeffi- 
cient of  y^  in  (3)  disappear,  namely,  by  putting 

(4)  k  =  -lb. 

U  we  substitute  this  value  of  k  in  ( o),  this  equation  assumes 
the  form 

(5)  y^  +  P^  +  '1  =  0, 
where 

( f) )  p  =  c  -  y)^  anil  q  =  d  —  lhc+  .^^  h^. 

Thus,  any  cubic  equation  (2)  may  be  reduced  to  the  form  (5) 
by  making  the  transformation^ 

(7)  •'^  =  ^  -  i  f>. 

in  other  words  (see  Art.  02),  by  increasing  the  roots  of  (2) 
by  ^b.  Equation  (5)  is  usually  spoken  of  as  the  cubic  in 
its  reduced  form,  or  as  the  reduced  cubic. 


198  INTEGRAL   RATIONAL   FUNCTIONS         [Art.  121 

The  reduced  cubic  equation  (5)  may  be  solved  as  follows: 
We  may  introduce  two  unknowns,  u  and  w,  such  that  their 
sum  shall  be  a  root  of  the  cubic  equation  ;  that  is,  we  put 

(8)  y  =  u+  y, 
whence 

7/3  =  ^3  j^  3  ^^2^,  _j_  3  ^^2  _|_  y3  —  yZ  j^  ^,3  ^  2)  Uv(u  +  w). 

Since  ^  =  m  +  y  is  to  be  a  solution  of  (5)  we  must  have 

^3  j^  y3  _|_  3  uiiQu  _|_  11^  -^  .pQn,  -)_  ;)  j  +(2=0, 

or 

(9)  V?  +  v^+  (3  uv  +p')(u  +  v)  =  —  q. 

Any  two  numbers,  u  and  v,  which  satisfy  this  equation 
will  have  for  their  sum,  w  +  v,  a  root  of  the  cubic  equation 
(5).  And  conversely,  if  we  wish  u  +  v  to  be  a  root  of  (5), 
then  u  and  v  must  satisfy  (9).  But,  since  two  unknowns, 
such  as  u  and  y,  are  not  determined  uniquely  by  means  of  a 
single  equation,  we  have  the  right  to  impose  the  condition 
that  u  and  v  shall  satisfy  some  other  equation,  besides  satis- 
fying (9).  This  second  equation  ma}^  be  chosen  at  our 
pleasure,  provided  it  does  not  contradict  (9).  The  infinitely 
many  different  choices  which  we  can  make  of  this  second 
equation  correspond  to  the  infinitely  many  ways  in  which  a 
given  sum  y  may  be  split  up  into  a  sum  of  two  terms  u  +  v. 

In  choosing  a  second  equation  for  u  and  y,  we  are  guided, 
of  course,  by  the  desire  to  make  this  second  equation  as 
simple  as  possible,  and  to  have  it  aid  us  in  simplifying  our 
first  equation,  that  is  (9),  at  the  same  time.  This  is  accom- 
plished by  choosing 

(10)  ?>uv+p  =  0 

as  the  second  equation.     As  a  consequence  of  (10),  equation 

(9)  reduces  to 

u^  -{•  v^  =—  q. 

We  have  obtained  the  following  preliminary  result.     Ifu 
and  V  are  two  numbers  which  satisfy  the  two  equations 


Aim.  \-2\]  THE   CUBIC   EQUATION  199 

(11)  ?f"  -f  v^  =  —  <i  and  3  uv  =  —  2)i 

then  their  sum,  u  +  >'  =  y,  will  he  x  root  of  the  cubic  equation  (5). 

Let  us  solve  the  second  equation  of  (11)  for  v.,  and  sub- 
stitute the  resulting  value  in  the  first.     We  find 

27  n^  ^ 

or 

(12)  U^-\-qu^-  .,\p^  =  0. 

But  this  is  a  quadratic  equation  for  u^  and  may  be  solved 
for  u'^  by  the  formulae  (5)  of  Art.  68. 
We  find  in  this  way 

(13)  ,,3  =-iq±  V:K,  where  R  =  J.  P^  +  I  f- 

Let  us  choose  the  upper  sign  in  (13)  and  let  us  denote  by 


(14)  7/i=V-  .]  ry  +  VK 

one  of  the  three  cube  roots  of  —\q-\- Vi2,  the  real  one  if 
—  \  q  +  ^R  is  real.  The  other  two  cube  roots  of  —\  q+VR 
will  then  be 

(15)  i/g  =  (w?<p   )i^  =  (o'^Ui 

where  co  and  co^  are  the  two  complex  cube  roots  of  unity, 
namely 

(16)  a,  =  _  I  +  1  /V3,  6)2  =  _  .1  -  1  ?V3.      (See  Art.  118.) 

From  (11)  we  find,  corresponding  to  each  of  these  three 
values  of  u,  a  value  of  i\  such  that  the  sum  u  +  v  will  be  a 
root  of  (5).  In  particular  we  find,  from  the  second  equation 
of  (11), 

But  this  expression  may  be  simplified  by  multiplying  both 
numerator  and  denominator  by  v  —  ^  9'  —  y/R.  We  obtain 
in  this  way,  making  use  of  the  value  of  R  given  in  (13), 


200  INTEGRAL   RATIONAL   FUNCTIONS         [Art.  121 


_  —  jpV-  o  <y  —  V/? 

and  this  will  reduce  to 

(17)  v^=V-  lq--^M 


if  the  symbol  V  —  I  9  —  ^^  ^^  defined  to  be  that  one  of  the 
three  cube  roots  of  —  .|  q  —  Vi2  whose  product  with  u^  is 
exactly  equal  to  —  J />.  If  the  coefficients  of  (5)  are  real 
numbers  and  if  B  is  positive,  the  cube  roots  «j  and  Vj  may 
be  selected  as  the  real  cube  roots  of  —  I  q  +  ^B  and 
—  I  q  —  ^ B  respectively. 
Having  selected  our  cube  roots  in  this  way,  we  know  that 


(18)  ij^  =  n^  +  i\  =</ -lq  +  V72  +  V  - 1  7 - -s/B 

is  one  of  the  three  roots  of  the  cuhie  equation  (5). 

If  instead  we  choose  for  u  another  one  of  the  three  cube 
roots  of  —  I  q  +  Vi2,  say 

the  corresponding  value  v^  of  ?'  will,  according  to  (11),  be 
given  by 

—  —   P    —  _     P     —  1 

since    —p/^u-^  is  equal  to   v^      If    we    multiply   numerator 
and  denominator  of  this  last  fraction  by  w^  we  find 
or'  o 

since  ©^  =  1.     Therefore  a  second  root  of  the  cubic  equation 
(5)  will  be 

(19)  y^  =  u^  +  ^2  =  (u?/i  -I-  (o^Vy 
Similarly  we  find  the  third  root  to  be 

(20)  t/g    =    Wg    +    ?'3    =    C0%1    +    (OVy 


Art.  1-21]  THE   CUBIC    EQUATION  201 

The  three  formulas  (18),  (19),  and  (-0),  for  the  roots  of 
the  reduced  cubic  equation  (5),  are  usually  known  as 
Cardan's  formulas.  They  were  first  published  in  printed 
form  by  Cakdano  (1501-1576)  in  his  famous  treatise  called 
Ars  Magna^  in  1545.  Cardano  however  was  not  the  dis- 
coverer of  these  formulas.  They  were  first  found  by 
SciPiONE  DEL  Fekko,  professor  of  mathematics  at  Bologna 
from  1496-1526,  and  perhaps  independently  by  Tartaglia 
(1500-1557).  Cardano  lias  been  severely  condemned  for 
publishing  these  formulas,  which  were  obtained  by  him  from 
Tartaglia,  only  after  he  had  sworn  to  keep  them  secret. 
It  seems,  however,  that  there  were  circumstances,  brought  to 
light  only  recently,  which  tend  to  free  Cardano  from  tlie 
odium  of  perjury. 

Remark.  In  developing  our  fornu\las  we  cliose  the  nj)p«M-  sign  in  (l-i). 
We  niiglit  just  as  well  have  choseu  the  lower  sign.  It  is  now  clear  that 
nothing  essential  would  have  been  changed  if  we  had  made  this. other 
choice.  The  result  would  have  been  a  mere  change  in  notation.  The 
quantities  which  we  have  actually  denoted  by  u's  and  y's  would  instead 
have  been  denoted  by  f's  and  u's  respectively. 

EXERCISE  LI 
1.    Solve  the  cubic  equation  x^  —  6  x-  +  6  x  —  2  =  0. 
Solution.     This  equation  has  the  form  of  (2).  Art.  121,  and  we  have 

(1)  h  =-6,  c  =^  +  t},  d  =  -  2. 

According  to  (.")),  (6),  and  (7),  Art.  121,  the  reduced  cubic  will  be 

(2)  //='  +  p;/  +7  =  0 
wheie 

(8)      />=  +  «-  i(-(j)^==-0.  >,  =  -2-  i(-(!)6  +  v^:(- (;)«=- 6. 

r  =  //  +  2. 

From  these  values  and  (lo),  Art.  121,  we  find 

(I)        /,'=  ,V(-(;)M-  ](-fir=  1. 

The  Cardano  formulas  (18),  (Kl).  (20)  of  Art.  121  now  give  us 

(o)  (/j  =  >/T  +  v'2,  I/.,  =  (1) v4  +  (o'-y/2,    //g  =  a>'v^4  +  <dV2. 

According  to  (:5),  tlie  corresponding  roots  of  the  original  equation 
are  just  2  greater  than  these,  so  that  they  are 

(«)  2  +  v^l  +  V2,    2  +  ai\/i  +  (U-V2,    2  +  (x)-\/i  +  w\/2. 


202  INTEGRAL   RATIONAL   FUNCTIONS         [Art.  122 

Verify  these  results  by  direct  substitution,  and  wiite  out  the  last  two 
roots  in  more  detail  by  substituting  the  values  of  to  and  aS^  from  (16), 
Art.  121. 

Solve  the  following  cubic  equations  and  check  your  results  by 
substitution. 

2.  jfi  -'dx-2  =  0.  4.    x^  -Q  X  -  40  =  0. 

3.  x3  -  9  X  -  28  =  0.  5.    a;3  +  4  X-  +  4  X  +  3  =  0. 

122.  Discussion  of  the  roots.  If  p  and  q  are  real  numbers 
and 

(1)  ii=^hp'  +  \q' 

is  positive,  the  root 

Vi  =  "i  +  «'i 

is  real,  and  clearly  the  other  two  roots,  y^  and  y^  are  conju- 
gate complex  quantities. 

If  R  is  negative,  all  three  roots,  as  given  by  the  Cardano 
formulas,  seem  to  involve  imaginary  quantities.  One  can 
show  however  that,  in  this  case,  the  imaginary  terms  destroy 
each  other  and  that  all  three  roots  will  be  real.  A  full  dis- 
cussion of  this  case,  which  we  shall  omit  here,  leads  to  a 
very  convenient  solution  of  the  cubic  equation  by  means  of 
the  trigonometric  functions.  (See  Dickson's  Elementary 
Theory  of  Equations,  P'^ge  36.) 

The  solution  of  the  cubic  equation  by  means  of  Cardano's 
formulas  is  a  matter  of  great  importance  from  a  theoretical 
point  of  view.  For  purposes  of  actual  calculation,  however, 
tlie  methods  of  Chapter  IV  are  usually  much  to  be  preferred. 

123.  The  ratios  of  the  coefficients  of  the  general  cubic 
equation  expressed  in  terms  of  its  roots.  The  general  cubic 
equation 

Ax^  -{■  Bx^ -\-  Cx ->r  D  =  ^ 

may  be  written  in  the  form 

(1)  a;3  4-^:r2  +  -^.r+^=0. 

AAA 

If  2:j,  x^-,  Xq  are  the  roots  of  this  equation,  x  —  x^,  x  —  x^i 
and  X  —  x^  must  be  factors  of  the  cubic  function  which  con- 


Aim.  128]       COEFFICIENTS   IN   TERMS   OF    ROOTS  203 

stitutes  the  left  member  of  (1).  Consequently  this  left 
member  is  divisible,  without  remainder,  by  the  product 
(a;  —  a-j) (a;  —  oTg) (a:  —  3-3) .  Since  this  product  is  itself  a 
cubic  function  of  a;,  the  quotient  must  be  independent  of  a;, 
that  is,  the  quotient  must  be  a  constant.  Moreover,  since 
the  coefficient  of  a-^  in  (1)  is  equal  to  unity,  and  the  coeffi- 
cient of  2^  in  the  expanded  product  (a;  —  t^^{x  —  X2)(x  —  3^3) 
is  also  equal  to  unity,  the  constant  quotient  just  mentioned 
must  be  equal  to  1.  Therefore  we  must  have,  for  all  values 
of  X, 

.^3  +  _a:2  +  ^a:  +  ^  =  (x  -  x^)(ix  -  x,;^)(x  -  a-g) 

But  if  two  such  functions  are  equal  to  each  other  for  all 
values  of  a:,  the  coefficients  of  like  powers  of  x  in  the  two 
members  must  be  equal  (See  Art.  126,  Theorem  F),  that  is, 

_  =  -  (a'l  +  :r3  +  .rg), 

C 
(2)  -  =  x^x^  +  a-ga-i  +  x^x^, 

—  —  —  :i^x^2y 

Formulas  (2)  sJio^v  how  to  calculate  the  ratios  of  the  coeffi- 
cients of  a  cubic  equation  whose  three  roots  a-^,  a-g,  x^  are  given. 
The  coefficients  of  one  such  cubic  equation  are  found  by 
putting  A  equal  to 'unity.  Formulas  (2)  were  apparently 
first  noticed  by  Cardano  in  1545. 

We  found,  in  Art.  68,  similar  relations  between  the  coeffi- 
cients and  roots  of  a  quadratic,  and  we  shall  see  before  long 
how  these  relations  may  be  generalized  so  as  to  apply  to  an 
equation  of  the  n\\\  order. 

This  generalization  was  accomplished  by  the  great  French  mathe- 
matician ViETE,  usually  known  as  Vikta  (1540-1608),  in  1559,  but  only 
for  the  case  of  positive  roots.  The  Dutch  matlieniatician  Girard 
(159(X?-1632)  finally  dropped  this  unnecessary  restriction  in  1G29. 


204  INTEGRAL    RATIONAL    FLWCTIOXS         [Airr.  li^i 

124.    The  equation  of  the  fourth  order.      We  niiiy  write  any 
(■(luatiou  of  the  fourth  order  in  the  form 

(1 )  .T*  +  Aj^  +  B.r'  +  Cx  +  i>  =  0, 

and,  by  means  of  a  transformation,  analogous  to  (7)  of 
Art.  121,  namely 

(2)  x=i/-\A, 

reduce  it  to  an  equation  of  the  simpler  form 

(3)  f  +  qf  +  ri/  +  s  =  0, 

in  which  there  is  no  term  involving  i/^. 

We  shall  attempt  to  solve  (o)  by  a  method  similar  to  that 
which  was  used  in  solving  the  cubic.      We  shall  put 

(4)  ^  =  u  +  V  +  u\ 

thus  equating  ?/  to  a  sum  of  three  terms  (as  yet  unknown), 
whereas  in  the  case  of  the  cubic  we  equated  y  to  a  sum  of 
two  terms. 

We  find  from  (4) 

y^  =  li?  +  ?'2  -f-  ^2  _j_  0(inv  +  WU  +  ?/t') 

t/4  =  (  M^  +  ?'2  ^_  w'2  )2+  4(  ?(2  4-  ,.2  +  >r^)(V7V  +  WU  +  M?'.) 
+  4(  r2w2  4-  «,2i^2  _^  ^j2,,2)  4.  X  uf.a,(u  +  c  +  w). 

The  expression  (4)  will  be  a  root  of  (3)  if  and  only  if  tlie 
following  equation,  obtained  by  substituting  these  values  in 
(3),  is  satisfied  : 

(  ^2  ^   ,,2  +  ,^,2^2^  4(  ,^,2,^^,2  +  ^^,2^f2  ^  ,,2^,2  )  ^  ,^(  ,/2  +  ,.2  +  ^^,2)  ^  g 

(5)  +  ( ?'?^  +  WU  +  ur)[4{  11^  +  /'2  +  w')  +  i'  '/] 
+  («+?'  +  /r)(  S  /^/vr  +  r)  =  0. 

Now  we  may  satisfy  this  equation,  by  a  proper  choice  of 
M,  i\  and  w,  in  infinitely  many  ways.  The  simplest  way  con- 
sists in  making  the  last  two  lines  of  (5)  disappear,  by  im- 
posing tlie  two  conditions 

, ,,.  ?<2  _|_  ^,2  _|_  ^^,2  __  _  1  q^ 

(6)  - 

uviv  =  —  ^  r,  • 


AuT.  124]     THE    EQUATION   OF   THE    FOURTH    ORDER       205 

on  the  three  numbers  u,  v,  and  w,  and  tlieu  as  a  third  condi- 
tion, the  equation  to  which  (5)  reduces  as  a  consequence  of 
(6),  namely 

(7)  ( U2  +  y2  _,_  ^^2)2 ^  ^(^  ^,2^2  +  ^^,2,4^2  ^  ^^2^,2  ) 

+  ^(u2  +  v2  +  ?/.2  )  4.  .s-  =  0. 

But  this  hitter    equation    may  be    siniplitied    further,   on 
account  of  (6),  to 

lq2^  4(v2ufi  +  w^'X^  +  7/2,-2  )  _  1  ,y2  ^  .,  ^  0, 

or 

(8)  vhv^  +  whj?  +  uh^  =  Jg  (f/  -  4  s> 

Thus  the  sum 

y  =  n  +  V  +  w 

ivill  he  a  root  of  (3),  if  ?/,  v,  a«t?  z(>  <?rg  three  numbers  which 
satisfy  the  three  equations 

,Q.  w2  +  ^2  -^  ^^2  _  _  1.  q^      y^y^f,  =  _  1  ^^ 

^2?^  +  tvhl^  +  M2it2  =  _i-  (^cf  —  4  &•), 

whose  right  members  are  knoivn  in  terms  of  the  coefficients  of  (8). 
If  we  replace  the  second  equation  of  (9)  by 

(10)  u^v^U^=   6 T  **^' 

and  make  use  of  Art.  123,  we  see  that  we  can  now  write 
down  a  cubic  equation  of  which  w2,  v"^,  and  tv"^  shall  be  the 
three  roots.     In  fact  the  cubic  equation, 

(0  -  ?/2)(2  _  v2-)(2;  -  w2)  =  0 

or 

Z^—  (  w2  J^  1,2  ^  ^4»2^22  +  (  y2^^,2  _^  ^^,2j^2  _^  ?f2^,2)2  _  ^2y2^^2  _  Q^ 

in  which  2;  is  the  unknown  quantity,  lias  u^,  i»2,  w^  as  its  roots. 
But,  on  account  of  (9),  we  may  write  this  equation  as 
follows  : 

(11)  ^3  +  1  qz^  +  Jg  (ry2  _  4  s)z  -  ,\  r^  =  0, 

the  coefficients  of  this  cubic  being  known  in  terms  of  the 
coefficients  of  the  given  equation  (3). 


206  INTEGRAL   RATIONAL   FUNCTIONS         [Art.  124 

Let  2p  22,  ^3  be  the  roots  of  this  cubic  equation.  Then  we 
may  put 

and  therefore 

(12)  u  =  ±  Vgj,   V  =  ±  V22,  2i'  =  ±  V23. 

But  not  all  of  the  eight  possible  combinations  of  the  ambigu- 
ous signs  ±  are  permissible,  since  we  are  seeking  three  num- 
bers w,  y,  w,  which  satisfy  the  relations  (9).  The  first  and 
third  of  these  relations  will  be  satisfied  if  either  sign  be  given 
to  each  of  the  square  roots  which  occur  in  (12).  However, 
the  second  relation  of  (9),  namely 

uinv  =  —  |-  )\ 

may  be  used  to  determine  the  sign  of  w  after  the  signs  of  u 
and  V  have  been  chosen  arbitrarily. 

If  the  symbols  V^^,  'y/z^,  V^g  represent  square  roots  of 
2p  ^2'  %  including  a  +  or  —  sign,  the  signs  being  chosen  in 
such  a  way  as  to  make 

then  the  four  roots  of  the  quartic  equation  (3)  will  be 

^^=V2j+V22+V^, 

2/4  =  -  ^'^1  -  ^^H  +  ^H- 

The  cubic  equation  (11),  which,  according  to  our  method, 
must  be  solved  first  in  order  to  obtain  the  roots  of  the  quartic 
equation  (3),  is  called  the  resolvent  cubic.  Although  there 
are  many  other  methods  for  solving  a  quartic  equation,  they 
all  have  this  feature  in  common,  that  a  certain  cubic  equation 
must  be  solved  first  before  the  solution  of  the  quartic  can  be 
effected.  In  like  manner  we  saw  in  Art.  121  that  the  solu- 
tion of  a  cubic  requires  as  a  preliminary  the  solution  of  a 
certain  quadratic,  namely  the  quadratic  (12)  of  Art.  121,  for 


Art.  125]  EQUATIONS   OF   HIGHER   ORDER  207 

u^.  This  fact  manifested  itself  in  the  Cardano  formulas  by 
the  appearance  of  the  square  root  VJL 

Since  the  quantities  z^,  z^,  z^  are  roots  of  the  cubic  equa- 
tion (11),  we  could  express  their  values  in  terms  of  q,  r,  and 
s,  by  means  of  the  Cardano  formulas  and  thus  obtain  hnally 
the  expressions  for  the  four  roots  ^]«//2'//3'  ^i  "^  C'"^)  ^^  terms 
of  its  coefficients  i-y,  r,  and  s,  thus  completing;-  the  algebraic 
solution  of  the  quartic  equation.  It  is  clear,  however,  that 
the  resulting  expressions  will  be  very  complicated  and  not 
at  all  adapted  to  the  purposes  of  actual  calculation.* 

We  have  shown,  however,  that  the  roots  of  ani/  equation  of 
the  firsts  second,  third,  or  fourth  degree  may  he  expressed  in 
terms  of  the  coefficients  of  the  equation,  hy  means  of  formulas 
which  involve  only  a  finite  number  of  such  ojyerations  as  addi- 
tion, subtraction,  multiplication,  division,  and  the  extraction  of 
square  and  cube  roots. 

The  first  solution  of  the  quartic  equation  was  given  by 
Ferrari  (1522-1565),  a  student  of  Cardano's,  It  was 
published  in  15-15  by  Cardano  in  his  Ars  Magna.  The 
solution  which  we  have  given  is  essentially  the  same  as  the 
one  found  by  Euler  (1707-1783)  in  1732. 

EXERCISE    Lll 

Solve  the  following  equations  by  the  methods  of  this  article  : 

1.    .r^  -  -2  x^  -  8  .f  -  3  =  0.  2.  .r*  -  10  .r^  -  -"iO  j-  -  16  =  0. 

125.  The  equations  of  higher  order.  Having  found  an 
algebraic  solution  for  the  general  equations  of  order  1,  2,  3, 
and  4,  the  mathematicians  of  the  sixteenth,  seventeenth, 
and  eighteenth  centuries  turned  their  attention  to  the  gen- 
eral equation  of  the  lifth  order.  But  all  attempts  to  find  a 
similar  solution  for  the  general  equation  of  the  fifth  order 
failed.  In  1821,  the  great  Norwegian  genius  Niels  Hexruv 
Abel  (1802-1829)  showed  that  these  efforts  were  necessarily 
doomed  to  failure,  by  proving  that  it  is  impossible  to  express 

*  For  purposes  of  numerical  calculation,  the  methods  of  Chapter  IV  are,  in 
most  cases,  much  to  be  preferred. 


208  INTEGRAL   RATIONAL   FUNCTIONS         [Art.  126 

the  roots  of  a  general  quiutic  equation  in  terms  of  its  coeffi- 
cients by  means  of  a  finite  number  of  such  operations  as  addi- 
tion, subtraction,  multiplication,  division,  and  extraction  of 
roots.  Abel's  proof  was  long  and  complicated.  By  starting 
from  an  entirely  different  point  of  view,  Evariste  Galois 
(1811-1882)  introduced  new  methods  and  new  ideas,  which 
not  only  enabled  him  and  his  successors  to  materially  sim- 
plify the  proof  of  Abel's  theorem,  but  which  are  of  funda- 
mental importance  in  the  theory  of  equations  of  higher  de- 
gree. These  ideas  of  Galois,  leading  to  what  is  now  known 
as  tlie  Theory  of  Groups,  had  however  to  some  extent  been 
anticipated  by  his  great  compatriots  Lagrange  (1736-1813) 
and  Cauchy  (1789-1857).  It  is  fairly  evident  that  if  the 
equation  of  the  fifth  degree  cannot  be  solved  by  means  of 
the  operations  indicated  above,  then  the  same  thing  will  be 
true  of  equations  of  still  higher  degree. 

It  should  be  remembered,  however,  that  this  impossibility 
of  which  we  are  speaking  is  a  relative  one.  If  certain  oper- 
ations of  a  higher  character  are  introduced,  besides  the  mere 
extraction  of  roots,  the  equation  of  the  fifth  degree  can  be 
solved.  Thus  Hermite,  who  was  born  in  Paris  in  1822  and 
died  in  1901,  showed  that  the  quintic  equation  could  be  solved 
by  means  of  elliptic  functions. 

126.  The  fundamental  theorem  of  algebra.  We  saw  in  Art. 
55  that  an  equation  of  the  first  degree  always  has  one  solution. 
Admitting  complex  numbers  as  solutions  of  an  equation,  we 
proved  in  Art.  70  that  every  quadratic  equation  with  real  or 
complex  coefficients  has  two  roots,  real  or  complex,  distinct 
or  coincident.  The  case  of  coincident  roots  is  understood 
more  easily  if  we  express  this  theorem  in  a  different  form 
whicli,  on  account  of  the  factor  theorem  (Art.  81),  is  equiv- 
alent to  it;  namely,  every  quadratic  function  Ax^  +  Bx-\-  0 
may  be  expressed  as  a  product  of  two  linear  functions,  with 
real  or  complex  coefficients.  The  case  of  coincident  roots 
corresponds  to  the  case  where  these  two  linear  factors  are 
identical. 


Akt.  126]  THE  FUNDAMENTAL  THEOREM  OF  AL(JEBRA   209 

The  algebraic  solution  of  the  cubic  and  ([uartic  equations 
which  we  have  given,  always  gives  rise  to  real  or  complex 
numbers  as  roots  of  such  equations.  Again  we  mav  state 
the  results  thus  :  Every  cubic  function  As^  +  Bj'^  +  Ox  +  D 
may  be  expressed  as  a  product  of  three  linear  functions  with 
real  or  complex  coefficients.  These  three  linear  functions 
may  either  be  different  from  each  other,  or  else  two,  or  even 
all  three  of  them,  may  be  identical.  The  situation  is  similar 
in  the  case  of  a  quartic  function. 

Again,  we  saw  in  Art.  120  that  every  equation  of  the  form 
x"  —  a  =  0  has  n  distinct  complex  numbers  as  roots.  Accord- 
ing to  the  factor  theorem,  then,  x"  —  a  may  be  expressed  as  a 
product  of  71  linear  factors  witli  complex  coefficients. 

All  of  these  things  we  have  'ActnaWy  proved.  They  sug- 
gest that  the  following  theorem  is  probably  true. 

A.  Every  mteyral  rational  function  of  the  nth  degree  cari  he 
decomposed  into  a  product  of  n  factors  of  the  first  degree,  the 
coefficients  of  the  factors  being  real  or  complex  numbers. 

If  we  associate  with  each  of  these  factors  a  root  of  the  cor- 
responding equation,  one  root  for  each  linear  factor  whether 
the  factors  are  all  different  from  each  other  or  not,  we  may 
also  state  this  theorem  as  follows. 

B.  Every  equation  of  the  nth  degree  with  real  or  complex 
numbers  as  coefficients  has  precisely  n  roots,  each  of  ivhich  is  a 
real  or  complex  number. 

This  theorem  is  usually  known  as  the  fundamental  theorem  of  Algebra. 
We  have  actually  proved  it  for  n  =  1,  2,  8,  and  1.  TIh'  theorem  was  first 
formulated  by  Gikahd  (1590-1G32,  Dutch)  iu  his  Inrent'um  nouvelle  en 
ralgehre,  published  in  1029.  But  the  first  satisfactory  proof  of  the 
theorem  for  all  values  of  n  was  given  by  Gauss  iu  his  doctor's  disserta- 
tion of  1799.  All  of  the  proofs  which  had  been  attempted  before  then,  by 
d'Alembert  (1717-1783,  French),  by  Euler  (1707-1783,  Swiss),  and 
by  Lagrange  (1736-1813,  French  but  born  in  Italy),  were  inaccurate. 

As  we  have  indicated  in  tiie  preliminary  discussion,  the 
statement  A  is,  in  some  respects,  preferable  to  statement  B. 
There    are    actually  always    precisely   n  linear    factors,  but 


210  INTEGRAL   RATIONAL  FUNCTIONS         [Art.  126 

some  of  them  may  be  equal  to  each  other,  giving  rise  to 
factors  of  the  form  {x  —  a)^  (x  —  a)^,  etc.  In  such  cases 
the  corresponding  equation  seems  to  iiave  fewer  than  n  roots, 
and  statement  B  only  remains  true  as  a  consequence  of  the 
agreement  that  such  a  root  a  shall  be  counted  more  than 
once.  These  are  the  so-called  multiple  roots  of  the  equa- 
tion.    (See  Art.  109.) 

The  proof  of  the  fundamental  theorem  is  too  difficult  to 
be  given  here  completely.  However,  this  proof  is  composed 
of  two  parts,  a  difficult  and  an  easy  part.  The  difficult  part 
may  be  stated  as  follows  : 

C.    If  any  integral  rational  function 

/(.r)  =  ^.r"  +  Bx"-^  +  •  •  •  ^  Lx  +  M 

is  given,  whose  coefficients  A,  B,  ■■■.L,  M  are  any  real  or 
complex  numbers,  there  always  exists  at  least  one  real  or  com- 
plex number  a-j,  for  tvhich  the  function  f  (x)  assumes  the  value 
zero.  That  is,  the  equation  f  (x)  =  0  has  at  least  one  real  or 
complex  root  x^ 

If  we  grant  this  much,  we  can  show  at  once  (this  is  the 
easy  part  of  the  proof)  that  the  equation  has  exactly  n 
roots,  and  not  merely  a  single  one. 

For  let  rcj  be  the  one  root  of  /  (.r)  =  0  whose  existence  is 
assured  by  theorem  C.  Then  a;  —  a;^  is  a  factor  of  f  (x), 
and  we  shall  have 

(1)  /(^)  =  i^  -  ^O/iC^), 

where  f-^ipo)  is  an  integral  rational  function  of  degree  n  —1. 
According  to  theorem  C,  the  equation  f^i^x)  =  0  has  a  root  ; 
call  it  x^.     Then  x  —  x^is  di  factor  of /j(a;),  so  that 

(2)  f,(^x)  =  ix-x,y^{x\ 

where /2(a:)  is  of  degree  n  —  2.     From  (1)  and  (2)  we  find 

(3)  /  (x)  =  {x  -  .r  1  )(x-  x^^ (.-c) . 
As  we  continue  in  this  way  we  finally  find 

(4)  /(a;)  =(x-  x^){x  -x^)  ■■■  {^x  -  x^}A, 


Akt.  126]  THE  FUNDAMENTAL  THEOREM  OF  ALGEBRA  211 

where  A  does  not  contain  x  at  all,  so  that  A  is  a  constant. 
Thus  /(a^)  has  precisely  n  linear  factors  x  —x^,  •••,  x  —  x„. 
and  this  is  the  fundamental  theorem  in  the  form  as  given  in 
statement  A. 

The  formula  (4)  leads  to  an  im}3ortant  corollary.  It 
shows  that  /(.t)  will  become  equal  to  zero  for  x  =  x^^  for 
x=  x^,  •■■■,  for  X  =  Xn,  and  for  no  other  values  of  x,  unless 
A  should  happen  to  be  equal  to  zero.  Of  course,  if  ^  =  0, 
the  function /(a;)  will  be  equal  to  zero  for  all  values  of  x, 
and  all  of  the  coefficients  of /(a:)  in  its  expanded  form  would 
be  equal  to  zero.     Wo  may  formulate  this  as  follows : 

D.  If  an  integral  rational  function  of  the  nth  order  is  equal 
to  zero  for  more  than  n  distinct  values  of  x^  then  it  will  be  equal 
to  zero  for  all  values  of  x. 

If  an  equation  f(x')  =  0  holds  for  all  values  of  x,  it  is 
called  an  identity,  and  the  sign  =  is  frequently  used  to  indi- 
cate such  an  identical  equation.  For  the  sake  of  contrast 
an  equation  which  is  not  an  identity  is  frequently  called  a 
conditional  equation.  We  may  now  reformulate  D  as 
follows : 

E.  If  an  i7ite(/ral  rational  function  of  the  nth  order 

f  (x)  =  Ax''  +  Bx"-'^  +  ••  ■  +  Lx  -f  M 

is  equal  to  zero  for  more  than  n  distinct  values  of  x,  then  it  is 
equal  to  zero  identically^  and  we  must  have 

We  have  already  used  some  special  cases  of  this  theorem 
(see  Art.  123,  for  instance)  and  we  shall  soon  apply  it  again. 
Most  of  the  applications  will  present  themselves  in  the  fol- 
lowing form. 

F.  //  the  equation 

(5 )  Ax"  +  Bx"-' -{-... -\-Lx-\- M=  A'x^  +  5'.r"-^  +  •  •  •  +  L'r  +  M' 

is  an  identitij,  that  is,  if  it  is  satisfied  Iti/  more  titan  n,  and 
therefore  hi/  all  values  of  x,  then  we  may  conclude  that 
A  =  A!,  B=B\     ,L=L',  M=M'. 


212  INTEGRAL   RATIONAL   FUNCTIONS         [Art.  127 

For,  if  (5)  is  an  identity,  then 
(A  -  A')x^  -\-(B  -  B')x"-'  +  ••■  +  {L-L')x  +M-M'  =  0 
is  also  an  identity,  and  according  to  E,  J.  —  A\  B  —  B\  ••■. 
L  —  i',  and  M—  M'  must  all  be  equal  to  zero. 

EXERCISE  Llll 

Examine  the  following  equations.  Are  they  identities  or  conditional 
equations  V     Give  a  proof  for  your  answer  in  every  case. 

1.  x'-  -  2  X  +  5  =  0. 

2.  (x-  1)2  =  x^~  2x  +  1. 

3.  .5  x^  -  7  x^  +  8  .r  -  1  =  0. 

4.  x^  -  x{x  -  1)2+  (x  -  1)2  -  3  x2  +  3  X  =  1. 

5.  (x  -  2)  (x  -  3)  =  x2  -  f)  X  +  6. 

6.  x2  —  (a  +  h)x  +  ah  =  (x  —  a)(x  —  />). 

7.  What  value  must  k  have  in  order  that  x^  +  A:x  +  9  may  be  identi- 
cally equal  to  (x  +  3)^? 

8.  If  the  equation  x'^  +  px  -[■  q  =  {x  +  r^  is  an  identity,  what  rela- 
tions must  there  be  between/)  and  ?•,  q  and  r,  p  and  q'i  If  these  con- 
ditions are  satisfied,  what  relation  is  there  between  the  two  roots  of  the 
quadratic  equation  x'^+  px  -\-  q  —  i)"} 

9.  Let  r^  and  r„  be  the  roots  of  the  equation  x-  +  px  +  ^  =  0.  If  i\^ 
is  just  twice  1\,  what  relation  must  there  be  between  p  and  ^y  ? 

10.  If  all  three  roots  of  the  cubic  equation  x^  +  px"^  +  qx  +  r  —  Q  are 
equal  to  each  other,  what  relations  are  there  between  p,  q,  and  r? 

127.  Application  of  the  fundamental  theorem  to  functions 
with  real  coeflBcients.  If  the  coefficients,  A^  -S,  •••,  M^  of  an 
integral  rational  function 

(1)  / {x)  =  Ax^  4-  Bx""-^  +  ■■■  +Lx  +  M 

are  real  numbers,  the  values  of  the  function  which  corre- 
spond to  real  values  of  x  will  be  real.  To  an  imaginary  value 
of  a:,  such  as 

(2)  x=a  +  hi     h  ^  0, 

tliere  will  correspond  in  general  an  imaginary  value  of  the 
function,  say 

(3)  f{a  +  hi)==P+Qi, 


Akt.  1-27]     FUNCTIONS   WITH   REAL   COEFFICIENTS  213 

which  reduces  to  a  real  number,  if  and  only  if  Q  happens  to 
be  equal  to  zero. 

Let  us  now  consider  the  value 

(4)  X  =  a  —  hi, 

which  is  conjuf^ate  to  (2).      (See  Art.  34.)     We  shall  find 

(5)  f(^a-hi)  =  P-QU 

which  is  conjugate  to  (8).  To  prove  this  statement  observe 
first  that,  since  we  are  assuming  the  coefficients  of  (1)  to  be 
real,  the  symbol  i  enters  into  the  expression  (3)  oi  f  (^a  +hi) 
only  because  i  occurs  in  x=  a-{-  hi.  Since,  in  the  second 
place,  a  — hi  may  be  written  a  -(-^(— ^),  the  valueof /(«  — ii) 
may  be  obtained  from  that  oifQa  +  hi)  if  we  replace  i,  when- 
ever it  occurs  in  the  latter  expression,  by  —  i.  But  this  last 
statement  is  equivalent  to  (o). 

We  may  therefore  state  tlie  following  theorem :  An  integral 
rational  function  f  {x)  with  real  coefficients  assumes  conjugate 
complex  values  for  conjugate  complex  values  of  the  independent 
variable  x. 

If  rt  +  hi  is  a  root  of  the  equation /(.r)  =  0,  we  have 

(6)  f(a  +  hi)  =  P  +  Qi  =  0, 

an  equation  which  implies  two  others,  namely 

(7)  P  =  Q  =  0.  (See  Art.  25.) 
But  then  we  shall  also  have,  on  account  of  (5)  and  (7), 

(8)  f(a-hi)  =  P-  Qi  =  0, 

so  that  a  —  hi  is  also  a  root  of  the  equation  /  (a;)  =  0. 

Thus,  if  an  equation  with  real  coefficients  has  as  one  of 
its  roots  an  imaginarij  numher  a  +  hi,  the  conjugate  of  this 
number,  namely  a  —  hi,  will  also  he  a  root  of  the  same  equation.* 

*  III  formulating:  this  theorem  we  have  used  the  word  iniaf/inanj  rather  than 
the  word  compk'.r,  for  the  followiiis;  reason.  Accordintj  to  the  hest  usage,  the 
term  complex  number  is  used  for  all  numbers  of  the  form  a  +  hi  and  therefore  in- 
cludes in  particular  the  real  numbers,  namely,  if  ^  =.  0.  We  use  the  term  imagi- 
nary number  for  those  complex  numbers  which  are  not  real,  that  is,  for  those  for 
which  It  is  not  equal  to  zero.  The  above  theorem  is  true  for  all  complex  numbers, 
inchidinji  reals,  but  it  is  of  no  interest  in  the  latter  case  since  the  conjugate  of  a 
real  number  a  is  that  same  real  number. 


214  INTEGRAL   RATIONAL   FUNCTIONS         [Art.  127 

This  fact  is  often  expressed  as  follows.  Tlie  imaginary 
roots  of  an  equation  with  real  coefficients  occur  in  pairs. 

From  this  it  follows  at  once,  that  an  equation  with  real 
coefficients  either  has  no  imaginary  roots  at  all  or  else  an  even 
number  of  them.  It  cannot  have  an  odd  number  of  imaginary 
roots. 

Consequently,  an  equation  of  odd  degree.,  with  real  coeffi- 
cients, always  has  at  least  one  real  root. 

Let  a  +  hi  be  an  imaginary  root  of  the  equation  with  real 
coefficients, 

(9)  f(^x)  =  Ax""  +  5a;"-i  +  •  •  •  +  ice  +  i¥  =  0. 

Then  a  — hi  is  also  a  root  of  (9).  Consequently  x—  (^a-{-hi') 
and  a^  —  (a  —  5^')  are  factors  oif{x).  (See  Art.  84.)  But  we 
have 

[re  —  (a  +  hi)']  \^x  —  (a  —  hi')']  =■  (x  —  a  —  bi^Qx  —  a  -\-  bi^ 

=  (^x—  a)^+  b^. 

That  is,  these  two  conjugate  complex  linear  factors  of /(.r) 
combine  into  a  single  real  quadratic  factor.  If  we  combine 
this  result  with  the  fundamental  theorem,  we  obtain  the  fol- 
lowing theorem : 

An  integral  rational  fu7iction  with  real  coefficients  may  al- 
ways be  expressed  as  a  product  of  real  linear  and  real  quadratic 
factors. 

EXERCISE    LIV 

1.  Given  f(x)  =  2  a:2  -  3  a;  -  1.  Compute  /(I  +  i),  /(I  -  i)  ; 
f(2  +  3i),f(2-Si). 

2.  Given  f{x)  =  x^-2x^  +  3x-7.  Compute  f(i),  f(-i); 
/(-l  +  20,/(-l-20. 

3.  State  a  reason  for  the  fact  that  there  exists  no  cubic  function 
with  real  coefiicients  which  has  x  =  1,  x  =  1  +  i,  and  x  =  2  as  its  zeros. 
Find  a  cubic  function  with  imaginary  coefficients  which  has  these  three 
numbers  as  zeros. 

4.  A  cubic  equation,  with  real  coefficients,  has  the  roots  x  =  I  and 
X  =  2  +  8  /.     What  must  its  third  root  be?    Find  such  a  cubic  equation. 

5.  Find    a    cubic    equation    whose    roots    are   1,     —  I  +  ^  /Vy,    and 


Arts.  128.  120]    (JllAPH  IN  CASE  OF  DISTINCT  FACTORS     215 

6.  Find  a  (luartic  ecjuation  "whose  roots  are  1,  —  1,  /,  —  i. 

7.  Prove  that  a  cubic  equation,  with  real  coefficients,  always  has  at 
least  one  real  root.  May  it  have  two  and  only  two  real  roots?  (In  an- 
swering this  question  remember  that  a  double  root  counts  for  two  roots. 
See  Art.  126.) 

8.  How  many  imaginary  roots  may  an  equation  of  the  fifth  degree, 
with  real  coefficients,  have  ? 

9.  Discuss  examples  1-10  of  Exercise  XLTII  again,  making  use  of 
what  you  have  learned  in  the  meantime.  Obtain  all  of  the  information 
you  can  about  the  number  of  positive,  negative,  and  imaginary  roots  of 

these  equations. 

10.  Making  use  of  the  notions  permanence  and  complete  equation,  ex- 
plained in  Example  11  of  Exercise  XLIII,  prove  the  following  theorem : 
If  a  complete  equation  has  all  of  its  roots  real,  it  will  have  as  many  posi- 
tive roots  as  variations,  and  as  many  negative  I'oots  as  permanences. 

128.  Use  of  the  factored  form  of  /(jc)  in  plotting.  If  the 
linear  factors  of  a  function /(a;)  are  known,  it  is  very  easy  to 
draw  the  graph  of  the  function.  Its  intersections  with  the 
a;-axis  are  obtained  by  a  mere  inspection  of  the  factors. 
Moreover,  the  sign  of  each  factor,  and  consequently  the  sign 
of  f(x),  for  a  given  value  of  a;,  may  also  be  obtained  by 
inspection.  The  form  of  the  graph,  in  a  given  instance,  de- 
pends of  course  upon  the  nature  of  the  factors  of  f(x)- 
Consequently  we  are  led  to  distinguish  the  following  three 
cases. 

Case  I.  All  of  the  linear  factors  of  /(a;)  are  real  and 
distinct. 

Case  II.  All  of  the  linear  factors  of  /(a;)  are  real  but 
they  are  not  all  distinct. 

Case  III.     Some  of  the  linear  factors  of /(a,-)  are  imaginary. 

129.  Form  of  the  graph  in  the  case  of  real  and  distinct 

factors. 

Illustrative  Example.  Let/(jr)  =  (x  -  l)(x  -  2)(x  -  3).  Then 
the  graph  of  y  -  f{x)  will  intersect  the  a.-axis  in  the  three  points.  A,  B, 
and  C,  of  Fig.  55,  for  which  x  has  the  values  1,  2,  and  3  respectively. 
Let  us  divide  the  plane  into  four  regions  by  drawing  the  lines  AL,  BM, 
and  CN,  parallel  to  the  /y-axis  through  A,  B,  and  C  respectively. 


Fig.  55. 


INTEGRAL   RATIONAL   FUNCTIONS         [Art.  129 

Any  point   to  the   left  of   AL  has  an  abscissa 
less  than  unity.     For  such  a  point  we  shall  have 

x-KO,  a;-2<0,  x-3<0, 
and  consequently 

y=  (x  -  \)(x  -  2){x  -3)<0. 

Therefore  all  points  of  our  graph  to  the  left  of  AL 
have  negative  ordinates.  In  particular  we  find  for 
X  =  0,  y  =  (  —  1) (—  2) {  —  -i)  =  —  Q,  giving  the  point 
D  of  Fig.  55. 

The  abscissa  x  of  any  point  between  AL  and 
BM  satisfies  the  inequalities 

X  -  1  >  0,  X  -  2  <  0,  X  -  .}  <  0. 

Consequently,  we  shall  have  for  such  a  point 

y  =  (x  -  l)(x  -  2)(x  -  2)  >  0. 

That  is,  all  points  of  our  graph  which  lie  between  AL  and  BM  have  positive 
ordinates. 

In  particular,  we  find  for  x  —  3/2, 

x=3/2,  ^=  i(-  i)(_|)=  +  f, 
giving  the  point  marked  between  A  and  B. 

The  abscissa   of  any   point  between  BM  and    CN  satisfies  the  in- 
equalities 

X  -  1  >  0,  X  -  2  >  0,  X  -  3  <  0, 
making 

^  ^  =  (x-l)(x-2)(x-3)<0. 

Therefore,  all  points  of  our  graph  between  BM  and  CN  have  negative 
ordinates. 

In  particular  we  find,  for  x  =  5/2, 

^  =  l»  y  =  i(i)(~  2)  =  ~  f> 
giving  the  point  marked  between  B  and  C. 

The  abscissa  of  any  point  to  the  right  of  CN  satisfies  the  inequalities 

X  -  1  >  0,  .r  -  2  >  0,  X  -  3  >  0, 
making 

//=(x-l)(x-2)(x-3) 

positive.     Therefore,  all  points  of  our  graph  to  the  right  of  CN  have  positive 
ordinates. 

The  form  of  the  graph,  as  indicated  in  Fig.  55,  is  now  apparent. 
This  form  will  not  be  altered  very  essentially  if  we  replace 

(x-l)(x-2)Cx-3) 
by  /l(x-l)(x-2)(x-3), 

where  A  is  any  positive  or  negative  constant. 


Aim.  loO]    GRAPH  IN  THE  CASE  OF  REPEATED  FACTORS  217 


EXERCISE  LV 
Draw  the  graphs  of  the  following  functions. 


1. 

2. 

3. 

4. 

9. 
10. 
11. 


11=  i(.r-l)(x-2)(x-:5). 
.y=-(.r-l)(.r-L>)(x-;}). 

,,=  (,,  _  i)(,.  _2)(x-3)(x- 
.y:^(..-  \)(x~2){x-^){x- 


5.  y  =  x(.r-  1)(.?;  -  2). 

6.  y={x+  \)x{x-\). 

7.  >,=  (,-+l)(,,--l)(^-3). 

8.  y  =  {x-  l)(x-  :3)(x-o). 

4). 
4)(.i:-5). 


Describe  the  graph  of  /y  =  (x  —  «)(x  —  b){x  —  <•)  if   a,  h,  c  are 
any  three  distinct  real  numbers  arranged  in  ascending  order  of  magnitude. 

12.    Describe  the  graph  of  i/  =  (x—  rt,)(x  —  ^2)  ""  (^  —  ('u)  if  ('u  02  •••  a„ 
are  n  distinct  real  numbers  arranged  in  ascending  order  of  magnitude. 

130.   Form  of  the  graph  in  the  case  of  real  factors  some  of 
which  are  repeated. 

Illustkative  Example.  Let  f(x)  =  (x  —  l)'^(x  —  2).  Then  the 
graph  of  the  function  >/  =/(x)  has  the  points  A  and  B  of  Fig.  56  in 
common  with  the  x-axis,  since  ?/  =  0  for  x  =  1  and  for  ^y  j^  j,^ 
X  =  2.  But  this  time  the  curve  does  not  cross  the  x-axis 
at  A  ;  it  merely  touches  it.  We  draw  the  lines  AL  and 
BM  parallel  to  the  ^-axis.  For  all  points  to  the  left  of 
^L  we  have 

a;<  1,     X  -  1  <0,     X  -  2<0 

and  therefore  Fig.  56. 


(x  -  1)2  >  0,  X  -  2  <  0,  >j  =  {x  -  \y-{x  -  2)<  0. 

Thus  all  points  on  our  graph  to  the  left  of  AL  are  below  the  x-axis. 
For  every  point  between  .4  L  and  BM,  we  have 

X  -  1  >  0,         X  -  2  <  0. 

Since  (x  —  1)^  occurs  as  a  factor  of  f(x)  we  find  y  <0  for  x>  1  as  well 
as  for  x<l.  Therefore  all  points  on  our  graph  between  AL  and  BM 
are  again  below  the  x-axis.  Consequently  the  graph  passes  through  A 
but  does  not  cross  the  x-axis  at  this  point.  According  to  Rolle's  theorem 
(.\rt.  108)  it  touches  the  x-axis  at  .1.  In  fact  if  we  expand /(x)  and 
compute  its  derivative  /'  (x),  we  sliall  find  /'(I)  =  0,  so  tliat  the  tangent 
of  the  graph  at  A  is  the  .r-axis.     At  B  the  curve  crosses  the  x-axis. 

This  case,  illustrated  in  Fig.  56,  may  be  regarded  as 
resulting  from  the  case  illustrated  in  Fig.  55  by  allowing  the 
point    B   of   Fig,  55  to    approach    the  point  A  as  a    limit. 


218  INTEGRAL   RATIONAL   FUNCTIONS         [Art.  130 

From  this  point  of  view  we  may  think  of  the  curve  as  cross- 
ing the  2;-axis  at  A  (Fig.  56)  twice,  or  of  a  point  P  which 
moves  along  the  curve  as  crossing  the  a^-axis  at  A  and  immedi- 
ately crossing  back  again.  We  should  therefore  conclude 
that  a  curve  obtained  from  that  of  Fig.  55  by  allowing  all 
of  the  three  points  A,  B,  and  C  to  coincide  would  correspond 
to  the  graph  of  ^=(^x  —  1)^.  It  is  clear  from 
/  Fig.  57,  which  represents  the  graph  of  (a;  —  1)^, 

Ay  that  we  may  actually  think  of  it  in  this  way. 

This  form  of  the  graph  is  characteristic  of  such 
/  cases  in  which  a  linear  factor   occurs   cubed. 

The  point  A  is  called  a  point  of  inflection  of 

Fig.  57.  t  -n  i  i-  /■,• 

the  curve.  It  will  not  be  dimcult  now  to 
decide  how  the  curve  will  look  in  the  neighborhood  of 
a;  =  r,  if  x  —  r  is  n  quadruple  or  quintuple  factor  of /(a;). 

EXERCISE   LVI 
Draw  the  graphs  of  the  following  functions  : 

1.  ^  =  2(x  -  l)2(r  -  2).  -7.  y=(x-  iy(.r  -  2)(x  -  3). 

2.  y  =  l(x-  l)Xx  -  2).  8.   1/  =  (.r  -l)(x  -  2)-2(x  -  3). 

3.  ,j=-(.c-iy2(x-2).  9.  >,=(.r-\)(x-2)(x-Sy. 

4.  y  =  -2(x  -  iy^(x  -  2).  10.  y=(x-  2)^. 

5.  y=  (x  -  l)(x  -  2)2.  11.  y  =  -  (x  -  2f. 

6.  ?/  =  -(x- l)(z- 2)2.  12.  ?/  =  i(x- 2)3. 

13.  If  Tj,  ?•„,  •••,r„  are  distinct  real  numbers,  show  that  the  graph  of 
y  =  A(x  —  r^y^(x  —  r.,)(x  —  r.,)  •••  (x  —  r„)  crosses  the  x-axisat  the  points 
for  which  x  =  r.„  r,,  •••,;•„  and  that  it  passes  throughi  the  point  of  the 
a;-axis  for  which  .c  =  ;-j,  without  crossing.  Prove  that  the  curve  is  tan- 
gent to  the  X-axis  at  this  point. 

14.  Prove  that  the  graph  of 

y  =  A(x  -  ri)'5(x  -  ?-2)---(x  -  ;•„), 

where  r^,  r^--- i\  are  distinct  real  niunbers  is  tangent  to  the  x-axis  at 
X  =  Tj  and  also  crosses  the  x-axis  at  this  point,  as  in  the  case  illustrated 
in  Fig.  57. 

15.  What  can  you  say  about  the  nature  of  the  graph  of 

y  =  A  (x  -  rj)*(x  -  r^)  •••  (x  -  r„) 
at  the  point  of  the  x-axis  for  which  x  =  ?-,?     Distinguish  between  the 
cases  when  k  is  an  even  or  an  odd  integer. 


Arts.  l;il.  1:3-2]         ROOTS   AND   COEFFICIENTS  219 

131.  Form  of  the  graph  when  some  of  the  linear  factors  are 
imaginary.  If  f{.r)  is  an  integral  rational  function  with 
real  coeliicients,  its  imaginary  factors  occur  in  conjugate 
pairs,  and  each  of  tliese  pairs  gives  rise  to  a  real  quadratic 
factor  (Art.  127).  Moreover  the  quadratic  factors  obtained 
in  this  way  do  not  vanish  for  any  real  value  of  x.  Conse- 
quently, if  n  is  the  degree  of  /  {x)  and  if  /  (x-)  has  2  k 
imaginary  factors,  the  graph  oif{x)  will  intersect  the  a;-axis 
only  in  n—  2  k  points. 

EXERCISE  LVII 
Draw  the  graphs  of  the  following  functions : 

1.  y  =  x^  +  X  -\-  1.  6.   y  =  (x  -  l)(x2  +  1). 

2.  7j  =  3(x-^  +  x+1).  7.  y  =  (x  +  l)(x  -  l)(x2  +  1). 

3.  yz=-  :3(x-^  4  X  +  1).  8.  y  =  x«  -  1. 

4-  i/  =  K^'  +  ^  +  !)•  9-  y  =  (2  X  +  5)(a;2  -x  +  1). 

5.  y  =  (.r  -  1)(.>;2  +  X+1).  10.  y  =  (x-iy\x^  -x+1). 

132.  Relations  between  the  roots  and  the  coefficients  of  an 
algebraic  equation.  We  found  it  to  be  impossible  (see  Art. 
125)  to  find  a  simple  expression  for  the  roots  of  an  equation 
of  the  nth  order  in  terms  of  its  coefficients.  But  the  inverse 
problem,  to  express  the  coeiHcients  in  terms  of  the  roots,  may 
be  solved  with  ease. 

Let 

(1)  «(,:?;"  +  a^a;"-!  +  a^x''-'^  +  ••  •  +  a^-iX  +  «„  =  0 

be  any  given  equation.     If  we  divide  both  members  by  Uq,  we 
obtain  the  equivalent  equation 

(2) .     f(x')  =  x^  +  p^x--'^  +  />2.r"-2  +  •  •  •  +  p„_ix  +  p,  =  0, 

where  we  have  put 

(3)  ;.,  =  ^,    p2  =  «2,...;,„_^  =  ^,  ^„  =  £-". 

Let  a:j,  x^-,  ■•■  Xn  be  the  roots  of  this  equation.  According 
to  the  factor  theorem  x  —  x^,  x  —  x^,--x  —  Xn  will  then  be 
factors  of /(a;),  that  is,  of  the  left  member  of  (2).  (See 
Art.  84.)     According  to  the  fundamental  theorem  of  Algebra 


220  INTEGRAL   RATIONAL   FUNCTIONS         [Art.  132 

(Art.  126),  f  i^x)  has  exactly  n  linear  factors.  Therefore 
/  (x)  has  no  other  factor  depending  upon  x ;  that  is,  the 
quotient 

(4)  -^^ =  A 

(X  —  X^^(X  —  Xc^^)  ■■■  {x-x^} 

is  independent  of  x,  so  that  yl  is  a  constant.     The  equation 

(4)  or  the  equivalent  equation 

(5)  /  (X)  =X^+  p^X^  -1  +  7V~^  +    •  •  ■    +  Pn-l^  +  Pn 

=  A(^x  -  x^) {X  -  Xg)  '■■  (x-  .r„) 

must  hold  for  all  values  of  x,  that  is,  it  must  be  an  identity. 
Therefore  the  following  equation,  obtained  from  (5)  by 
multiplying  together  the  factors  x  —  x^,  x  —  x^,  x  —  x^,  etc.,  of 
the  right  member, 

(6)  a:"  +  |>i2-"'-i  +  ^^2:c"-2  +  •  •  •  +  Pn-i-t'  +  Pn 

=  A[x^-{x^  +  x^-\-  ■■■  +.r„).r"-i 

-f-  (^XyV^  +  X^X^  +  •  •  •+  -^'w— i-*'n  )*i'      "         •  •  ■    ±  X^2'2  ■  ■  •  Xj^j , 

must  also  be  an  identity. 

But  according  to  Art.  126,  Theorem  F,  an  identity  of  this 
form  can  subsist  only  if  the  corresponding  powers  of  x  in  the 
two  members  of  the  equation  have  the  same  coefficients. 
The  coefficients  of  x"  on  the  right  and  left  members  are  A  and 
1  respectively.  Therefore  A  must  be  equal  to  unity,  and  we 
shall  have  besides,  comparing  the  coefficients  of  corresponding 
powers  of  x  in  the  two  members, 

i^i  =  ^^  = -(2-1  +  ^2+  •••  +^n), 

P2  ^        ^^  "I"  \-^\-^2  '   "^V^S    '     ' ' '     I    "^y^n  ~i    •^2'^S    '      ' ' '     '    "^2*^" 


"r    ■  ■  ■    "r  '-^n—l-^nyt 


a 


O)  PS=^=-  C-^'rVs  +  •''r'V^4  +    ■  •  •    +  ^n-2«n-ia^n). 


<l 


a 


P4 —  —  ~r  C-' r' 2"' 3*^  4     '      '"    "t"  ■^n-3'^n-2*^n— l-^n)> 


a 


Aim.  l:!:5]  SYMMETRIC    FUNCTIONS  221 

where  the  +  or  —  sign  is  to  be  used  in  the  last  equation  ac- 
cording as  n  is  even  or  odd. 

We  have  already  obtained  these  relations  in  the  special 
cases  when  w  =  2  or  3  (Art.  68  and  Art.  123).  The  gen- 
eral relations  (7)  were  tirst  discovered  .by  Girard.  (See 
the  historical  note,  Art.  123.) 

133.  Symmetric  functions.  The  expressions  which  occur 
in  the  riglit  members  of  (7  j,  Art.  132,  have  important  prop- 
erties. Except  for  sign,  p^  is  equal  to  the  sum  of  all  of  the 
roots,  and  therefore  jOj  does  not  change  its  value  if  any  two 
of  these  roots  are  interchanged.  Again  p^  is  equal  to  a  sum, 
each  term  of  which  is  equal  to  a  product  of  two  of  the  roots, 
and  all  possible  products  of  this  sort  occur  in  p^.  Therefore 
p^  will  not  be  changed  if  any  two  of  the  roots  are  inter- 
changed. The  same  thing  is  true  of  jOg,  p^,  •■•,Pn-  We  shall 
be  able  to  express  these  facts  very  concisely  with  the  aid  of 
the  following  definition. 

An  expression  S(x-^,  x.^,  •  •  •, .r„),  involving  n  letters  x^,  a^g,  •  •  •»  a;„ 
is  said  to  be  a  symmetric  function  of  x^,  x^-,  ■■■-,  x^  if  its  value  is 
left  unaltered  ivhen  any  two  of  these  letters  are  interchanged. 

Thus,  xi^  +  x.^  +  x^  is  a  symmetric  function  of  x,,  Xj,  x^.  But  x-^ 
—  2  Xo^  +  3  x.i-  is  not  symmetric.  For  if  we  interchange  x■^^  and  x^,  this 
becomes  x^^  —  2x^-+'^  x^-,  which  is  not  the  same  as  the  original  expression. 

We  may  now  express  a  part  of  what  is  involved  in  equa- 
tions (7)  of  Art.  132  as  follows. 

The  coefficients  of  an  algebraic  equation  of  the  form 

X""  +  p^X''-'^  -\- p^X""'"^  +    ■••    +Pn-l^+Pn=^ 

are  symmetric  functions  of  its  roots. 

The  expressions  Pj,  p^^,  ■•■  Pn  ^re  more  specifically  called  the 
fundamental  symmetric  functions  on  account  of  the  following 
important  theorem  due  to  Newton. 

Any  integral  rational  function  ofx-^^  x^^  ••o  ^n  which  is  sym- 
metric can  be  expressed  as  an  integral  rational  function  of  the 
n  fundamental  symmetric  functions  p^,  p-^  ■■■  Pn- 


222  INTEGRAL   RATIONAL   FUNCTIONS         [Art.  134 

No  proof  of  this  theorem  will  be  attempted  in  this  book. 

It  is  clear  from  what  we  have  said  that  the  fundameiital 
symmetric  functions  of  the  roots  of  a  given  equation  may  be  ob- 
tained from  that  equation  by  inspection^  although  the  indi- 
vidual roots  themselves  may  be  entirely  unknown.  This 
remark  is  very  important  in  many  applications. 

EXERCISE   LVIII 

1.  Solve  the  equation  x^  —  Q  x-  +  26  a;  —  24  =  0,  making  use  of  the  in- 
formation that  the  three  roots  of  the  equation  form  an  arithmetical  pro- 
gression. 

2.  Find  the  roots  of  x^  —  8  a;^  +  5  a;  +  14  =  0,  making  use  of  the  fact 
that  the  sum  of  two  of  the  roots  is  equal  to  9. 

3-  Given  a  cubic  equation  x^  +  p^x'^  +  p.^x  +  p..  =  0.  Find  a  formula 
for  ./.'i'-  +  x/  +  x^  in  terms  of  the  coefficients. 

4.  If  7^,  =  0,  what  can  we  say  about  the  roots  of  the  corresponding 
equation  ? 

5.  Solve  x^  —  8  x^  +  5  a;  +  .50  =  0,  being  given  that  two  of  the  roots  are 
equal. 

134.    Vanishing  and  infinite  roots.     If  the  equation 

(1)  /'  (a;)  =  a^^x^  +  a^x"-'^  +  •  •  •  +  (tn-iX  -^a^  =  0 

has  one  of  its  roots  equal  to  zero,/(i')  must  reduce  to  zero 
when  we  put  x  =  0.  Therefore  «„  must  be  equal  to  zero. 
The  same  thing  results  from  the  factor  theorem.  For  if 
a;  =  0  is  a  root  of  /(.r)  =  0,  x  must  be  a  factor  of  fii-c)-  If 
x=  0  is  a  multiple  root^  taking  the  place  of  k  simple  roots^  x'' 
must  be  a  factor  off(x^;  that  is,  the  coefficients  of  the  k  terms 
of  lotvest  order  in  (1)  must  be  eqtial  to  zero. 

Tlie  same  result  might  have  been  obtained  from  equations 
(7)  of  Art.  132. 

Instead  of  considering  a  single  equation  of  form  (1),  let 
us  now  consider  a  whole  chain  of  such  equations.  Let  the 
coefhcients  a^  a^---  a^  be  the  same  for  all  of  the  equations  of 
the  chain,  so  that  the  individual  equations  will  differ  from 
each  other  only  in  regard  to  the  value  of  «q.  Let  us  assume 
further  that  a„  is  different  from  zero,  and  that  a^  approaches 


Art.  l:3i]         VANISHING   AND   INFINITE  ROOTS  223 

zero  as  we  pass  from  the  first  equation  to  the  second,  from 
the  second  to  the  third,  and  so  on. 

An  example  will  make  clear  what  these  assumptions  mean.     Let  the 
equations  all  be  quadratics,  and  let 

1)  1*5^-^4  5x  -7  =  0, 

2)  T^  x--^  +  5  X  -  7  =  0, 

3)  ToVff  ^■'■^  + -"'a:  -  7  =  0, 

k)  _Lj;2+  5x  -  7   =  0, 

be  the  first  k  equations  of  the  chain.     For  the  A-th  equation  we  have 
1  -  7 

The  values  of  a^  and  a^  are  the  same  for  all  equations  of  the  chain ;  a„,  in 
our  case  a„,  is  different  from  zero  since  a^  —  —  7.  Finally  Oq  is  different 
from  equation  to  equation  and  approaches  zero  as  a  limit  as  k-  gi'ows  be- 
yond bound. 

According  to  (7),  Art.  132,  we  have 


if  a^j,  2^2'  ••■'  ^n  ai'e  the  roots  of  equation  (1).  As  a^ 
approaches  zero,  the  quotient  aJaQ  will  grow  beyond 
bound,  since  a^  is  different  from  zero  by  hypothesis.  Con- 
sequently at  least  one  of  the  roots  a^i.a^g,  •••■tX^,  whose  prod- 
uct, according  to  (2),  is  equal  to  a„/«o,  must  grow  beyond 
bound. 

So  far  in  this  discussion  we  have  assumed  a^^O.  If  in- 
stead a„  =  0,  the  equation  (1)  has  at  least  one  root  equal  to 
zero  ;  that  is,  f(x)  has  some  power  of  x  as  a  factor.  If 
we  divide  f{x)  by  this  power  of  .r  and  apply  our  argument 
to  the  resulting  equation,  which  has  no  vanishing  roots,  we 
conclude  that  at  least  one  of  its  roots,  which  is  also  a  root  of 
the  original  equation,  must  grow  beyond  all  bound  when  a^ 
approaches  zero  as  a  limit. 

Therefore  we  have  the  following  result.  //'  the  coefficient 
a^,  of  the  highest  power  of  x  in  an  equation  of  the  form 


224  INTEGRAL   RATIONAL   FUNCTIONS         [Art.  134 

(1)  ((qX"  +  a^x"-'^  +  •'•  +  a„_i^'  +  «„  =  0 

be  regarded  as  a  variable  which  approaches  the  limit  zero,  then 
at  least  one  of  the  roots  of  the  equation  will  grow  beyond  bound. 

This  is  sometimes  expressed  by  saying  that  the  equation 
has  an  infinite  root. 

We  may  state  the  following  more  general  and  more  pre- 
cise result. 

If  the  first  k  coefficients  of  (1)  be  regarded  as  variables  which 
simidtaneously  approach  the  limit  zero,  but  if  the  (k  +  V)th 
coefficient  of  (1)  remains  finite.,  then  precisely  k  of  the  roots  of 
(1)  ivill  grow  beyond  bound. 

In  this  case  the  equation  is  said  to  have  k  infinite  roots. 
To  prove  this  last  theorem  we  might  proceed  as  in  the  case 
of  one  infinite  root,  making  use  of  equations  (7)  of  Art.  132. 
But  it  is  easier  to  reduce  the  case  of  infinite  roots  to  the  case 
of  vanishing  roots  by  means  of  the  transformation 

(3)  a;=-,  y  =  -. 

y  ^ 

If  we  make  this  substitution  in  (1),  we  find,  after  clearing 
of  fractions, 

(4)  a^  +  a^y  +  a^"^  +  •  •  •  +  a„_^y«-'^  +  a^y""  =  0. 

From  (3)  it  is  clear  that  the  roots  ?/p  y^..  •••  y^  of  (4)  and  the 
roots  a^j,  2^21  •••  ^n  of  (1)  ^^e  so  related  that  we  may  put 

1  1  1 

X-^  X,^  X^ 

Consequently,  if  k  roots  of  (1),  say  x^,  x^,  •••  x,^,  grow  beyond 
bound,  then  k  roots  of  (4),  namely  y^,  y^,  ■•■  3/^,  will  approach 
the  limit  zero.  But  we  have  seen  that  (4)  will  have  exactly 
k  vanishing  roots,  if  and  only  if  the  left  member  of  (4)  con- 
tains y'''  as  a  factor,  that  is,  if  a^  =  a^=  •••  =  a^._|  =  0,  %^  0. 
Therefore  we  have  proved  our  theorem  about  the  infinite 
roots  of  (1).  This  tlieorem  finds  an  important  application 
in  analytic  geometry  in  the  theory  of  asymptotes. 


Art.  134]        VANISHING   AND    INFINITE    ROOTS  225 


EXERCISE  LIX 

1.  By  actually   solving   the   quadratic   equation    iu   the   illustrative 
example  of  Art.  IM,  namely 

—  x-  +  5a;-7  =  0, 
10* 

prove  directly  that  one  of  its  roots  tends  to  become  infinite  with  growing 
values  of  k,  while  the  other  one  tends  toward  the  value  7/5. 

2.  What  relations  must  exist  between  a,  h,  k;  and  in,  in  order  that  the 
following  equation 

(??i%'^  —  b-)x-  +  2  kiiKi'-j-  +  (k-  +  h-)n-  =  0 

may  have  one  infinite  root?  two  infinite  roots  ?  Discuss  these  relations 
under  the  assumption  that  a  and  h  are  different  from  zero,  while  k  and 
m  may  be  either  zero  or  diffei'ent  from  zero. 


CHAPTER  VI 

FRACTIONAL  RATIONAL  FUNCTIONS 

135.  Definition  of  a  rational  function.  If  we  divide  one 
integral  rational  function  by  another,  two  cases  may  present 
themselves  ;  the  division  may  be  exact  or  not.  In  the 
former  case  the  quotient  is  again  an  integral  rational  func- 
tion, and  may  therefore  be  studied  by  the  methods  of 
Chapters  IV  and  V.  In  the  latter  case,  the  quotient  is  a 
new  kind  of  a  function,  known  as  a  fractional  rational 
function.  The  fractional  and  integral  rational  functions 
together  constitute  the  class  of  rational  functions. 

Every  function  which  can  he  expressed  either  as  an  integral 
rational  function,  or  else  as  a  quotient  of  two  integral  rational 
functions,  is  called  a  rational  function. 

For  the  sake  of  brevity  we  shall  often  speak  of  a  fractional 
rational  function  as  a  rational  fraction.  The  two  integral 
rational  functions,  of  which  the  rational  fraction  is  the 
quotient,  may  be  called  its  numerator  and  denominator. 

Thus  ~   is  a  rational  fraction. 

x^  —  i 

x^  —  3  X  +  2  is  its  numerator,  and  x^  —  4  is  its  denominator. 

136.  Proper  and  improper  rational  fractions.  We  shall  say 
that  a  rational  fraction  is  a  proper  fraction  if  its  numerator 
is  of  lower  degree  than  its  denominator  ;  we  shall  call  it  an 
improper  fraction  if  the  degree  of  the  numerator  is  as  high 
as,  or  higher  than,  that  of  the  denominator.  We  observe  im- 
mediately that  a  rational  function  which  is  represented  hy  an 
improper  fraction  may  he  expressed  as  the  sum  of  an  integral 
rational  function  and  a  proper  fraction. 

226 


Art.  l:5(ij      1>R0PER   AND   IMPROPER   FRACTIONS  227 

Thus  we  have,  for  instance, 


X  —  1  X—  1       X  —  5  X  —  5 

The  integral  part  of  a  rational  fnnction  is  obtained  as  the 
quotient  in  the  process  of  dividing  the  numerator  by  the 
denominator.  The  remainder  obtained  in  this  division,  if 
there  is  a  remainder,  will  be  the  numerator  of  the  proper 
fraction  which  must  be  added  to  the  integral  part  in  order 
that  the  sum  may  be  equal  to  the  given  improper  fraction.  If 
the  remainder  is  zero,  the  given  rational  fraction  is  really  an  in- 
tegral rational  function  ;   it  is  fractional  only  in  appearance. 

The  notions,  proper  and  improper  fraction,  are  very  closely  related  to 
the  corresponding  notions  in  arithmetic.  In  arithmetic  d/D  is  said  to 
be  a  projier  fraction  if  d  is  less  than  D,  an  improper  fraction  if  d  is  not 
less  than  D.  Thus  the  notion  "  lower  degree  "  takes  the  place  of  the 
similar  notion  "  less  than,"  as  we  extend  the  terminology  of  arithmetic 
to  the  field  of  rational  functions. 

In  one  very  important  respect  the  theorems  about  rational 
functions  differ  essentially  from  the  corresponding  theorems 
about  the  fractions  of  arithmetic.  The  sum  of  two  proper 
arithmetical  fractions  may  be  an  impi'oper  fraction.  Tims 
1/2  +  3/4  =  5/4.  But  for  rational  functions  we  have  the 
theorem:  the  sum  of  two  rational  functions^  each  of  which  is 
a  proper  fraction,  is  always  again  a  proper  fraction. 

For  let  -p  r  ^  -f  r  \ 

be  two  proper  rational  fractions,  so  that  /j(.?')  and  ./^^-^O  ^^^ 
of  lower  degrees  than  ^/^(.r)   and  g^i-O  i"cspectively.       We 

This  is  again  a  rational  fraction  (compare  the  definition  in 
Art.  135),  and  the  numerator  is  of  lower  degree  than  the 
denominator. 

We  shall  soon  n)ake  an  important  application  of  this 
theorem. 


228  FRACTIONAL   RATIONAL   FUNCTIONS       [Art.  137 

137.  Reduction  of  a  rational  function  to  its  lowest  terms. 
Let  /.^  X 

be  a  rational  function  of  x,  f{x)  and  ^(a;)  being  integral 
rational  functions  of  degree  m  and  n  respectively.  Accord- 
ing to  the  fundamental  theorem  (Art.  126),  f^x)  may  be 
written  as  a  product  of  the  form 

/(a;)  =  A(x  -  aj^^)(x  -  a^)  ...  (x  -  a^), 

where  ^  is  a  constant,  and  where  a^,  a^^  •••  a^  are  m  numbers 
whicli  may  be  real  or  imaginary,  all  distinct,  or  some  or  all 
of  them  equal  to  each  other.     Similarly  we  shall  have 

[/(x-)  =  B(x  -  h^X^  -  h^y-.(x  -  6„), 

and  therefore 

From  this  expression  we  may  conclude  that  R(x^  will 
become  equal  to  zero  when  x  assumes  any  one  of  the  m  values 
aj,  ^2'  ■■■'  ^mi  unless  the  same  value  of  x  should  also  cause  the 
denominator  of  (2)  to  vanish.  The  fraction  R(x)  would 
not  be  defined  for  such  a  value  of  x,  since  it  would  then  as- 
sume the  form  0-^0,  which  is  meaningless.  (See  Art.  21.) 
But  the  denominator  of  (2)  can  vanish  only  when  x  becomes 
equal  to  one  of  the  values  h^^  h^^  ■•■  h^.  Consetiuently,  the 
rational  fraction  R(x)  can  assume  the  indeterminate  form.  0/0 
only  if  its  numerator  and  denominator,  have  a  factor  in  common 
ivhich  contains  x. 

Thus,  if  J,,.  ^  2(a:  +  5)(x  +  3)(.r-l) 

^  ^  (x  +  7)(x  +  3) 

we  find  r. 

^        ^      0 
We  may  divide  both  terms  of  a  fraction  by  the  same  number  (provided 
that  the  divisor  is  different  from  zero)  without  changing  the  value  of  the 
fraction.     Now  a;  +  3  is  different  from  zero  whenever  x  is  not  e(]ual  to 
—  3.     Therefore,  the  above  fraction  /?(x),  and  the  simpler  function 


Art.  137]  REDUCTION   TO   LOWEST   TERMS  229 

X  +  7 
are  equal   to  each    other  for  all  values  of  x  except  for  x  =  —  3.     For 
X  =  —  3  these  functions  are  not  equal.     In  fact  ii'(—  3)  has  no  meaning, 

whereas  R,(-  3)  =  ^'"^"^^  =  -  4. 
4 

The  illustration  just  given  suggests  the  following  defini- 
tion : 

//'  the  numerator  and  denominator  of  a  rational  fraction 
have  a  common  factor  containing  x^  the  rational  function  is  not 
in  its  lowest  terms.  To  reduce  it  to  its  lowest  terms  we  divide 
both  uuinerator  arid  denominator  by  their  highest  common 
factor. 

If  the  rational  function  Rix)  is  not  in  its  lowest  terras, 
and  if  R^Qc)  is  the  function  obtained  from  R{x)  by  reduc- 
ing to  lowest  terms,  we  shall  have 

R^x)  =  R^(x) 

for  all  values  of  x  excepting  those  for  which  R(x)  is  not 
defined  at  all ;  namely,  those  which  give  rise  to  a  meaning- 
less expression  of  the  form  0/0  for  RQx'). 

The  reduction  of  a  rational  function  to  its  lowest  terms  is 
easily  accomplished  when  the  linear  factors  of  numerator 
and  denominator  are  all  known.  But  ordinarily  this  is  not 
the  case.  However,  it  is  not  necessary  to  knoiv  all  of  the 
linear  factors  of  the  numerator  and  denominator  hi  order  to 
accomplish  this  reduction.  It  suffices  to  know  their  highest 
common  factor,  and  this  may  ahvays  be  obtained  by  the  method 
of  successive  division  ivhich  is  essentially  the  same  as  the  pro- 
cess for  finding  the  greatest  common  divisor  of  two  integers. 
(See  Art.  5.) 

EXERCISE  LX 

Reduce  the  following  improper  fractions  to  the  form  of  an  integral 
rational  function  plus  a  proper  fraction : 

^     x^-'x"^  +  X  -\  g    x*  -  1 


X-  -f  a:  +  1 
X'  -  X  +  1 
X-  +  X  +  1 


x  +  1 

X*  -  3  x3  +  2  x2  +  X  - 

1 

X8  +   X2  +  X  +   1 

230      FRACTIONAL   RATIONAL   FUNCTIONS       [Arts.  138,  139 

5.  Find  the  highest  common  factor  of  the  numerator  and  denominator 
of  each  of  the  fractions  in  Examples  1-4  and  use  the  result  to  decide  the 
question  whether  or  not  these  fractions  are  in  their  lowest  terms. 

Reduce  the  following  fractions  to  their  lowest  terms : 
g     (x~l)(x  +  2)  Q    3:2  -  5  X  +  4 

(j:  +  3)(x-1)'  ■         x-1 

rj    (x^  ^  X  +  l)(x-  1)  3    x^-  13r  +  42 

x*^  -  I  '     x2  -  7  X  +  6 

138.  Zeros  of  a  rational  function.     Let 

be  a  rational  function  in  its  lowest  terms.  If  a^  a^,  ■■•  a^ 
are  the  roots  of  the  equation  /(.r)  =  0,  then  R(x)  will  be 
equal  to  zero  when  x  assumes  one  of  these  values.  We  shall 
therefore  speak  of  these  values  a^,  a^^  •■•  a^  as  the  zeros 
of  RQx).  We  should  perhaps,  more  specifically,  call  them 
the  finite  zeros  of  R(x^t  since  7t(.r),  if  it  is  a  proper  fraction, 
approaches  zero  as  a  limit  when  x  grows  beyond  bound,  as 
we  shall  see  later.  Of  course  some  or  all  of  tha  zeros  of 
RQc)  inay  be  imaginary.  Moreover,  several  of  the  roots 
ftj,  ^21  •■■  *m  o^  tliG  equation  f  (x)  =  0  may  coincide,  that  is, 
/(x)  may  have  a  repeated  factor.  If  /(a;)  has  (x  —  rt^)** 
as  a  factor,  a^  is  called  a  multiple  zero  of  f(x'),  and  the 
number  r  is  called  the  multiplicity  of  this  multiple  zero. 

139.  Poles  of  a  fractional  rational  function.     If  5^,  b^,  •••  b„ 

are  the  roots  of  the  equation  g(^x)  =  0,  the  function  R{x^  is 
not  defined  for  x  =  b^,  x  =  b2',  -  •  ■  x  =  h^.  We  should  have, 
for  instance,  n^i  ^       /-^i  n 

where /(Jj)  is  different  fi^om  zero,  since  the  fraction  is  sup- 
posed to  be  in  its  lowest  terms. 

Thus  the  function  ,,/  \      x  +  4 

X  —  1 
is  not  defined  for  x  =  1.     But  it  is  defined  for  all  other  values  of  x.     Let 
X  approach  1  as  a  limit.     Then  the  numerator  will  approach  the  limit  5, 


Art.  UO]       GRAPH    OF   A    RATIONAL   FUNCTION  231 

and  the  denominator,  x  —  1,  will  approach  the  limit  zero.  As  the 
denominator  becomes  smaller  and  smaller,  the  value  of  the  fraction  will 
grow  beyond  all  bounds.  We  express  this  by  saying  that  R(x)  becomes 
infinite  as  x  approaches  1  as  a  limit,  and  we  say  that  x  =  1  is  a  pole  of 
the  function  R(x). 

If  a  fractional  rational  function  is  written  in  its  lowest 
terms,  the  function  c/roivs  beyond  hound,  or  (^as  we  sat/}  becomes 
infinite,  when  x  approaches  as  a  limit  one  of  the  zeros  of  its 
denominator.  These  values  of  x  are  called  the  poles  of  the 
rational  function. 

These  poles  will  be  real  or  imaginary,  simple  or  multiple, 
according  as  the  zeros  of  the  denominator  are  real  or  imagi- 
nary, simple  or  multiple. 

140.  Graph  of  a  fractional  rational  function.  We  may 
make  a  graph  of  a  given  fractional  rational  function  RQc)  as 
in  the  case  of  an  integral  rational  function.     We  put 

y  =  RQx^), 

assume  arbitrary  values  for  x,  compute  the  corresponding 
values  of  y,  and  plot  the  points  whose  coordinates  are 
obtained  in  this  way.  But  in  doing  this  we  should  pay 
special  attention  to  the  poles  of  R(x},  if  i2(a:)  has  any  real 
poles. 

Thus,  if  we  wish  to  make  a  graph  of  the  rational  function 

1 

2/  =  -. 

X 

we  observe  at  once  that  x  =  0  is  a  pole,  the  only  one  in  this  case.  For 
X  =  0,  y  is  not  defined.  But  we  may  compute  values  of  y  corresponding 
to  values  of  x  which  are  very  close  to  zero.  We  find  the  values  indicated 
in  the  following  tables  : 

....1     BCD     E-     A'    B'     C     D'     E' - 

x-:5      2      1      I      \ 1  _  i  _l  _o   _3  ... 

>j.:\      \      1      2      4  -  -4  -2  -1  -i   -y. 

The  corresponding  points  are  plotted  in  Fig.  58  which  shows  clearly  the 
essential  properties  of  this  graph.  As  x  approaches  zero,  decreasing 
toward  zero  from  positive  values,  y  becomes  positively  infinite.  As  x 
approaches  zero  through  negative  values,  y  becomes  negatively  infinite. 


232 


FRACTIONAL   RATIONAL   FUNCTIONS      [Art.  140 


Fig.  58 


Thus,  there  is  a  discontinuity  (break)  in  the 
vicinity  of  x  =  0.  The  other  jjoints  of  the 
grapli  are  easily  supplied.  They  show  that 
y  approaches  zero  as  x  grows  beyond 
bound,  either  through  positive  or  negative 
values. 

The  curve  obtained  in  this  example 
belongs  to  the  class  of  curves  called  hyper- 
bolas. This  particular  hyperbola  is,  more 
specifically,  called  a  rectangular  hyperbola 
on  account  of  the  intimate  relation  which 
it  has  to  two  perpendicular  lines,  the 
X-axis  and  ^-axis,  which  are  known  as  its 
asymptotes. 


EXERCISE     LXI 
Draw  the  graphs  of  the  following  functions 


1-  y 


2. 

J. 

y  = 

X 

3. 

2 

X 

4. 

1 

y=       , 

X  —  1 

5 

o 

■'       x-1 

6. 

-  1 

y  = .,    -, 

X  —  1 

8. 

X  —  5 

y  =  — 7' 

x-1 

9. 

y  =  2  — -• 

X  —  1 

10. 

a  X  —  D 

x-1 

11. 

x-1 

^      (x--2)(x-3)- 

12. 

X2  +   X  +    1 

^  ~    x(x  -  1) 

m 

-  is  a  curve  of  the  same  gen- 

1 

13.  Show  that  the  graph  of  y  =  h  -V 

X  —  a 

eral  character  as  that  of  y  =  1/x,  whatever  the  values  of  a,  h,  and  m  may 
be.  That  is,  it  is  a  rectangular  hyperbola  whose  asymptotes  are  parallel 
to  the  X-axis  and  ?/-axis  and  which  intersect  at  the  point  x  =  a,  y  =  b. 

14.  Show  that  every  function  of  the  form 

y  —  ^^  +  y ^     (jo,  (/,  r,  s,  being  constants) 
rx  -f-  .s- 

which  does  not  reduce  to  a  constant  or  an  integral  linear  function,  may 
be  rewritten  in  tlie  form 

y  =  ''  + > 


and  that  its  graph  thei-efore  has  the  properties  indicated  in  Ex.  13. 


Art.  141]     FACTORED  FORM  OF  A  RATIONAL  FUNCTION   233 

141.  General  form  of  a  rational  function  in  terms  of  its 
zeros  and  poles.     Let 

be  a  rational  function  in  its  lowest  terms  ;  and  let  /(a:)  and 
^(a;)  be  of  degrees  m  and  w  respectively.  Then  we  may 
write 

f{x)  =  A{x~a^){x-a^)  ■■•  {x -a^), 

g(x)  =B(x.-h^){x-K^  ...  ix-h„), 
where  A  and  B  are  constants,  and  consequently 

^  ^  ^  ^  ix-b,)ix-b,)...(x-b^) 

where  k  =  A/B  is  a  constant. 

Formula  (1)  (/ives  an  explicit  expression  for  the  most  (jeneral 
rational  function  which  has  the  values  a^,  a^.  •••  a^  as  zeros,  and 
the  values  b^,  b,^,  •••  b^  as  poles. 

Since  we  have  assumed  R(x')  to  be  in  its  lowest  terms, 
each  of  the  numbers  a^,  •  ■  •  a^  will  be  different  from  each  of 
the  quantities  b^,  b^  ■■•  b^. 

If  several  of  the  a's  are  identical,  say  a-^  =  a^  =  ••■  —  a^, 
then  aj  is  called  a  zero  of  multiplicity  r  (Art.  138),  and 
the  numerator  of  R{x)  has  (x  —  a^y  as  a  factor.  If  several 
of  the  5's  are  identical,  say  b^  =  b.^  =  ■■■  =  />„  then  b^  is 
called  pole  of  multiplicity  s,  and  the  denominator  of  i?(.?') 
has  (a;  —  b^y  as  a  factor. 

Thus,  the  must  general  rational  fraction  which  has  a^  as  a 
zero  of  multiplicity  r^  a^  as  a  zero  of  multiplicity  rg,  and  so 
on,  will  be 

(o\  ji(^\  =  k(.^-  ^O^'C^  -  ^i)"'  •  •  •  (^  -  ^m)^» 

^"^  ^  (x-  b^y^(x  -  ^2)'^  ■■■  {x-  b„yn 

When  a  rational  function  is  expressed  in  tlie  form  (1)  or 
(2),  we  shall  say  that  it  is  written  in  its  factored  form. 


234  FRACTIONAL   RATIONAL   FUNCTIONS      [Art.  142 

EXERCISE  LXll 

Write  down  the  most  general  rational  function  which  has  the  follow- 
ing numbers  as  zeros  and  poles  : 

1.  Simple  zeros  for  x  =  1,  2,  3 ;  simple  poles  for  x  =  4,  5,  6. 

2.  Simple  zeros  for  x=  -1,  0,  +  1 ;  simple  poles  for  x=  -  2,  +2,  +3. 

3.  Simple  zeros  for  a;  =  1,  a  zero  of  multiplicity  3  for  a;  =  2,  a  pole  of 
multiplicity  2  for  x  =  3,  and  a  simple  pole  for  x  =  4. 

142.   Partial  fractions.     Let 

(1)  ^(0  =4^ 

be  a  proper  rational  fraction  in  its  lowest  terms.  Then /(a:) 
will  be  of  lower  degree  than  g^x}.  If  g(^x)  is  of  degree  n, 
let  x  —  x^,  X  —x^,  ■■■  X  —  Xr,  be  the  n  linear  factors  of  g(x), 
and  let  us  assume  that  all  of  these  factors  are  distinct. 
Since  R{x^  is  in  its  lowest  terms,  none  of  these  factors  will 
be  factors  oif(x),  and  consequently,  x-^,  x^,  ■■■  x^  will  be  poles 
of  R(x). 

We  may  then  write  R{x~)  as  follows  : 

(2)  M(x)  =  -^ ^,     ^ 7 T^' 

(x-Xi){x-x^)  ■■■  {X-Xr,} 

where  the  numerator  of  i2(2;)  is  at  most  of  degree  w  —  1, 
since  we  have  assumed  B(^x)  to  be  a  proper  fraction.  This 
expression  contains  precisely  n  constants  a^,  rtj,  •  ■  •  a„_j  which 
may  have  any  value  whatever,  and  it  is  the  most  general 
expression  of  its  kind.  Therefore,  the  most  general  proper 
rational  fraction  ivhich  has  the  values  x  =  x-^,  x  =  x^,  •■•  x  =  .r„ 
as  distinct  poles^  contains  n  arbitrary  coefficients  a^,  rt^, a^^  ■•• 

Let  us  now  consider  the  sum 
(8)  _^i]_  +  ^2_+  ...  4--^, 

/y»    /y  /y    ,y>  nf»     __^     nm 

where  J.^,  A.^,-'  '"  ^n  are  constants. 

This  sum  is  a  rational  function  of  x ;  its  poles  are  the  same 
as  those  of  R(x)^  and  since  each  term  of   (3)  is  a  propef 


Art.  142]      PARTIAL   FRACTIONS   FOR   SIMPLE   POLES      235 

fraction,  their  sum  is  a  proper  fraction.  (See  Art.  136.) 
Therefore  tliis  sum  may  be  rewritten  in  the  form  (2),  by 
reducing  the  several  fractions  in  (3)  to  the  common  denomi- 
nator (^x  —  x^(x  —  j'^)  •••  {x  —  Xn)  and  adding.  Moreover 
the'n  coefficients  A^  A^-,  ••■  Anin  (3)  can  be  determined  in 
such  a  way  as  to  make  the  sum  (3)  exactly  equal  to  any 
given  expression  of  the  form  (2). 

The  following  example  will  illustrate  this  statement  and  show,  at  the 
same  time,  how  these  coefficients  Ai  ■••  .1,,  may  be  determined. 
Let 

^'  ^'       (x-l)(x-2)(x-3) 

This  is  a  proper  fraction,  and  it  is  in  its  lowest  terms;  for  the  numerator 
does  not  vanish  when  we  put  x  equal  to  1,  2,  or  3.  We  therefore  attempt 
to  determine  the  coefficients  A,  B,  C,  in  such  a  way  as  to  make  the  sum 

(•'">)  ^  +  -^+-^,-^(^). 

X  —  1      X  —  2      X  —  i 

or,  what  amounts  to  the  same  thing,  so  as  to  make 

.Q.  A  (X  -  2)(x  -  3)  +  Bjx  -  l)(x  -  3)  +  C(x  -  1) (x  -  2)  _  p .  . 

^^  ■      (x-l)(x-2)(x-.3)                                -''^^^' 

or 

.-X  A (x-^-  .5  X  +  6)  +  ^(x-^-  4 X  +  3)  +  C{x^  -  3  x  +  2)  ^  „.  . 

^^  (x-l)(x-2)(x-3)                                    ^^^- 

The  denominator  of  -ff(x),  as  given  by  (4),  and  of  the  left  member  of 

(7)  are  exactly  the  same.  Consequently  the  fractions  (4)  and  (7)  will 
be  equal  for  all  values  of  x  (identically  equal),  if  and  only  if  their 
numerators  are  identically  equal,  that  is,  if  and  only  if 

(.4  +  B+  C)x-2+  (-  5^-4J5-3C)x+6^+35+2  C=2x2-r)x  +  7. 

According  to  Art.  126,  Theorem  F,  this  will  be  so  if  and  only  if 

A  +  B  +  C^  2, 

(8)  -.5.4  -45-3  C==- 5, 

6  /I  +  3  5  +  2  C  =  7. 

The  solution  of  (8)  gives  A  =  2,  B  =  —  5,  C  =  +  5.  Consequently  we 
have,  from  (4)  and  (5),  the  desired  result,  namely 

-.gx  2  X-  -  a  X  -I-  7  _  _2 5_  5 

^^  (x- l)(x-2^(x-3)      x-1      x-2      x-3" 

An  easier  way  of  obtaining  the  same  result,  avoiding  the  solution  of 
system  (8),  is  as  follows.  If  (6)  and  (4)  are  identically  equal,  we  must 
have 


236  FRACTIONAL   RATIONAL    FUNCTIONS      [Art.  142 

(10)  A(x  -  2)(x  -  8)  +  B{x  -  l){x  -  3)  +  C(x  -  l)(x  -  2) 

=  2  a:'-2  -  5  a;  +  7 
for  all  values  of  x.     For  x  =  I,  (10)  gives  us 

A(-  1)(-  2)  =  2  -  5  +7,  whence  A  =  2. 

Similarly  we  find  from  (10)  for  a:  =  2, 

5(  +  1)  (  _  1)  =  2  •  4  -  5  .  2  +  7,  whence  B  =-5; 

and  for  x  =  '■'> 

C(+  2)(+  1)=  2  •  9  -  5  .  3  +  7,  whence  C  =  +  5. 

The  second  method  indicated  in  this  example  is  especially 
convenient,  and  has  the  advantage  of  being  explicitly  appli- 
cable to  the  general  case,  thus  enabling  us  to  prove  that  the 
coefficients  vlj,  A^,  ■•■  A^  in  (3)  can  always  be  determined 
in  such  a  way  as  to  make  the  sum  (3)  equal  to  any  given  ex- 
pression of  the  form  (2).  We  shall  refrain  from  actually 
writing  out  the  formulae,  but  we  shall  state  the  resulting 
theorem. 

Let  Ii{x)  he  a  proper  rational  fraction  in  its  lowest  terms, 
whose  poles  are  all  distinct,  so  that  the  denominator  of  R(x) 
has  no  repeated  factor.  Let  x^,  x^,  ■■■  x^he  the  poles  of  R{x). 
Then  R(x^  may  he  ivritten  in  the  form 

(11)  R{x~)  =  -^^  +  -^a_  +  . . .  +  -Ar^_. 

When  a  rational  function  is  expressed  in  this  way,  it  is 
said  to  be  resolved  into  a  sum  of  simple  partial  fractions. 

The  metliod  which  we  used  for  resolving  RQr)  into  a  sum 
of  partial  fractions  is  called  the  method  of  undetermined 
coefficients.  It  is  cliaracteristic  of  this  method  that  we 
assumed  an  expression  with  certain  unknown  coefficients 
(undetermined  at  the  time),  and  that  we  found  the  values  of 
these  coefficients  subsequently  by  comparing  the  resulting 
expressions  with  certain  others  which  were  known. 

In  applying  the  above  theorem,  w%  must  not  forget  that 
i2(a;)  is  assumed  to  be  a  proper  fraction.  If  R{x^  is  not  a 
proper  fraction,  we  must  first  reduce  it  to  a  sum  of  an  in- 
tegral rational  function  and  a  proper  fraction.      (See  Art. 


Art.  14:3]     PARTIAL  FRACTIONS  FOR  MULTIPLE  POLES     237 

13G.)  We  may  then  apply  the  theorem  to  the  hitter  part  of 
R(x).  Again,  we  assumed  also  that  R(^x)  is  in  its  lowest 
terms.  If  it  is  not,  we  must  reduce  it  to  its  lowest  terms 
before  attempting  to  resolve  it  into  partial  fractions.  Finally 
we  assumed  the  poles  of  Rijc)  to  be  simple  poles,  so  that  the 
linear  factors  of  the  denominator  are  all  distinct.  We  did 
not  assume  these  linear  factors  to  be  real.  The  tlieorem 
applies  equally  well  to  the  case  of  real  or  complex  linear 
factors,  provided  tliat  no  two  of  them  are  equal.  In  practice, 
however,  the  formula  (11)  is  usually  applied  only  to  the  case 
of  real  and  distinct  factors. 


EXERCISE     LXIII 

Express  the  following  rational  functions  as  sums  of  simple  partial 
fractions  : 


2. 


(x+l)(x-l)(x-2)  x^-x 

143.  Resolution  into  partial  fractions,  when  the  poles  are 
not  simple.  If  R{x)  is  a  proper  rational  fraction  in  its 
lowest  terms,  but  if  some  of  the  factors  of  the  denominator 
are  repeated,  the  expression  of  R(x)  as  a  sum  of  partial 
fractions  will  be  somewhat  different  from  that  given  in  Art. 
142.     Let 

(1)  R(^x-)  =  ^^^^ 


x2  +  x  -  3 

^     1  -  x  +  G  x'- 

(•^•-l)(^-2)(x-3) 

X  —  x^ 

2  a-2  -  3  X  +  5 

5          1 

(.c  +  2)(x-3)(x-6) 

x^  -  a2 

2  x2  -  7  .7:  +  3 

g     1  +  X  -  6  x2 

{^x  -  a)\x  —  hy(x  —  cy 


be  such  a  proper  fraction,  where  the  integers,  r,  8,  t,  etc.,  in- 
dicate how  many  times  x  —  a,  x  —  b,  etc.,  are  repeated  as  fac- 
tors of  the  denominator.  Tlien  a  is  called  a  pole  of  R(x)  of 
multiplicity  r;  b\s  called  a  pole  of  multiplicity  s;  and  so  on. 
(Seo  Art.  139.)     If  the  degree  of  the  denominator  be  still 


238  FRACTIONAL   RATIONAL   FUNCTIONS       [Art.  143 

denoted  by  w,  we  shall  have 

(2)  r+s  +  t+  ■■■  =n, 
and  the  numerator  must  be  of  the  form 

(3)  /  (x)  =  Oq  +  a^x  +  a^x^  +  •  ■  — t-  a„_i2:"-i 

not  involving  x",  2'"'*'i,  etc.,  since  i?(.r)  is  a  proper  fraction. 

The  most  general  proper  fraction  which  has  (a;  —  ay  as 
its  denominator  is  of  the  form 

^  I X  <^o  ~r  c-^x  -}-  C2X  ~r  •  •  •  ~l~  c^_-^x 

^^  ix-ay 

The  numerator  of  this  fraction,  which  is  arranged  according 
to  ascending  powers  of  x,  may  be  rewritten  as  a  polynomial 
arranged  according  to  ascending  powers  of  x  —  a.     For  we 

have 

X  =  a  -{-{x  —  a), 

x^=  a^  -\-  2  a(^x  —  a)  +  (2:  —  a)^, 
etc.  etc. 

Consequently  the  numerator  of  (4)  may  also  be  written  in 
the  form 

A,  +  A,^^(x  -  a)  +  ^,_2(.r  -  a)^  +  •  •  ■  +  A^{x  -  ay-\ 

so  that  (4)  becomes 

(J)\  ^r I Ai=i !-•••+  2 |_    ^1    . 

(x  —  ay      {x  —  ay~^  {x  —  ay      X  —  a 

Thus,  the  most  general  proper  fraction  which  has  (^x  —  ay  as 
its  denominator  may  be  replaced  by  a  sum  of  the  form  (5), 
where  the  r  coefficients  A^,  A^,  ■■■  A^  are  arbitrary  numbers. 
Similarly  the  most  general  proper  fraction  which  has 
(x  —  by  as  its  denominator  may  be  replaced  by  the  sum 


(^x -  by    {x -  by-^  (x-bf    x-b' 

and  so  on. 


Art.  143]     PARTIAL  FRACTIONS  FOR  MULTIPLE  POLES      239 

We  see  that  the  sum 

Ar I        ^r-i         I    ...    I      ^1 

{x  —  ay      (x  —  ay~^  X  —  a 

(6)    + — ^ +       ^'-1 — +••    -f-^^ 

+  __ili +  llAiJ +   ...   +_kl_ 

^  (.i:  _  e)'  ^  (.r  -  c)'-l  ^  ^^:  -  c 

+ 

will  be  equal  to  a  proper  fraction,  whose  denominator 
(x  —  ay(x  —  hy(x  —  cy  ■■■  is  the  same  as  tliat  of  i2(.r),  and 
that  the  undetermined  coefficients  :  A-^,  A.^^  ■•■^  A^;  B^,  ■,  B,; 
Cj,  ■••,  Of,  ■■■  which  occur  in  (6)  are  precisely 

r  +  s  +  r  -I-  •  ■  •  =  n 

in  number,  on  account  of  (2).  But  the  numerator  (3)  of 
R{x)  contains  precisely  n  coefficients  which  we  regard  as 
given.  We  may  therefore  expect  that  it  will  be  possible  to 
determine  the  n  coefficients:  A^,  •••,-4;.;  jBj,  ,  B^;  (7j,  •••, 
Ct,  ■  •-,  of  (6)  in  such  a  way  as  to  make  the  sum  (6)  exactly 
equal  to  R(pc).  We  shall  not  attempt  to  prove  that  this  can 
actually  be  done,  but  leave  it  to  the  student  to  verify  this 
fact  in  the  following  examples. 

EXERCISE    LXIV 

O  r  -I-  ,5 

1.    Resolve — ^^—^ into  partial  fractions. 

(,_1)3(.,._3) 

Hint.     The  theory  of  Art.  143  suggests  that  we  shall  put 

•2  X  +  5                   '  1        ^       ^'         ,       g      ,      D 
~~  +  ~. ~rr^  T r  H — 


(.f  -  l)3(x  -  3)       {X  -  \y      (.1-  -  1)-      X  -\      .1-3 
Reduce  the  right  member  to  a  single  fraction  with  the  denominator 
(.r  —  l)3(.i-  —  3)  and  compare  numerators. 

Express  the  following  functions  as  sums  of  simple  partial  fractions  : 

2       x  +  1  4     6  x8  -  8  a:''  -  4  a:  +  1 

■    {x-\y'  '  x%x-\y 

^         x+\  5        .S  x3  -  f)  x^  -  2  X  -  1 


x{x  -  \y^  (X  +  \)\x  -  l)2(x  -  2) 


240  FRACTIONAL   RATIONAL   FUNCTIONS       [Art.  144 

144.  Modified  form  of  the  partial  fractions  in  the  case  of  im- 
aginary poles.  Tlie  developments  of  Arts.  142  and  148  are 
applicable  whether  the  poles  are  real  or  imaginary.  In 
the  latter  case,  however,  the  partial  fractions  obtained  in  this 
way  will  also  be  imaginary,  and  it  is  usually  convenient,  al- 
though not  strictly  necessary,  to  avoid  the  introduction  of 
imaginary  elements  when  the  function  under  discussion  is  a 
real  function. 

Suppose  then  that  R(x)  is  a  real  rational  function,  in  its 
lowest  terms,  and  a  proper  fraction.  By  this  we  mean  that 
all  of  the  coefficients  of  R(x)  are  real,  so  that  R{x)  assumes 
real  values  whenever  x  is  real.  The  poles  of  Jl{.r)^  which 
are  the  zeros  of  the  denominator  of  RQx).  will  then  either  be 
real  or  conjugate  complex.  (See  Art.  34.)  In  other  words, 
the  denominator  may  be  regarded  as  a  product  of  real  linear 
and  real  quadratic  factors.  (See  Art.  127.)  Let  x^+jjx+q 
be  one  of  these  real  quadratic  factors  which  cannot  be  fac- 
tored into  two  real  linear  factors.  The  most  general  proper 
fraction  which  has  x!^ -{-  px -\-  q  as  its  denominator  will  have 
the  form 

n\  Ax-\-  B 

x^  +  px  -\-  q 

and  there  must  be  a  term  of  this  form  among  the  partial 
proper  fractions  whose  sum  is  equal  to  R{x).  This  one  real 
term  takes  the  place  of  the  two  terms  with  conjugate  imag- 
inary linear  denominators,  which  would  arise  from  tlie  two 
imaginary  linear  factors  of  x'^  +  px  -f  q  if  we  were  to  apply 
the  method  of  Art.  142. 

If  the  quadratic  factor  x"^  +  px  -\-  q  is  repeated,  the  corre- 
sponding terms  of  the  expression  R(x)  as  a  sum  of  partial 
fractions  may  be  taken  in  th«!  form 

(2)  A,x  +  B,        ^      A_yr  +  B,_^      ^  I    --^r^  +  ^i 

(x^  +  px  +  qy      (x^  +  px  -f-  qy-"^  x^  -\-  px  +  q 

The  form  (•_*)  of  the  partial  fraction  development  is  suggested  by  the 
following  argument.  There  must  be  in  this  development  a  proper  frac- 
tion with  (x'^  +  px  +  qy  as  denominator.     Since  {x^  +  jyx  +  (/)'•  is  of  de- 


Art.  145]       FRACTIONAL    RATIONAL   EQUATIONS  241 

gree  2r  in  x,  the  most  general  numerator  which  a  proper  fraction  with 
this  denominator  can  have  is  of  the  form 

Hut  it  is  not  hard  to  see  that  such  an  expression  as  (3)  may  be  rewritten 
in  the  form 

(4)  (A,x  +  B,)(x'  +  px  +  f,y-^  +  (A.,x  +  B„)(x^+px  +  «?)-2  +  ... 

+  (/l,.ix  +  Br_i)(x-  +  px  +  y)  +  A,x  +  B„ 
and  if  we  divide  (4)  by  (x-  +  px  +  (/)•■  we  obtain  (2). 

EXERCISE    LXV 

1.    Resolve  ; into  a  sum  of  simple  partial  fractious. 

(X  -  iy\x'  +1) 

Hint.  Here  the  factor  (x  —  1)-  is  a  repeated  real  linear  fat-tor,  and 
X-  +  1  is  a  quadratic  factor  whose  linear  factors  are  not  real.  Tiiis  leads 
u«toput  ^•2_4^^5      ^       A  B         Cx+D 

(X  -   1)^(X2  +    1)  {x  -    1)2         X  ~    1  X^  +   1 

Reduce  the  right  member  to  a  single  fraction  with  the  denominator 
(x  —  l)"^(.f-  +  1)  and  compare  numerators. 

Express  the  following  functions  as  sums  of  partial  fractions: 

2    4 4     6  x3  +  2  x'^  +  2  X  -  2 

(x-  l)(xH  X  +  1)'  ■  x*-l    . 

„                  9-2x  -  5x2-4x+6 

3.    • ••  5.- 


(x  +  2)  (x2  -  2  X  +  5)  (x2-  X  +  l)2(x  -  3) 

145.  Fractional  rational  equations.  An  equation  all  of 
whose  terms  are  rational  functions  of  the  unknown  qnantiti/  is 
called  a  rational  equation.  If  at  least  one  of  the  terms  of  the 
equation  is  a  fractional  function,  the  equation  is  a  fractional 
equation. 

If  there  are  any  terms  in  the  right-hand  member  of  the 
equation,  these  terms  may  be  transposed.  Consequently  any 
rational  equation  may  be  written  in  tlie  form 

(1)  B^ix)  +  ll,(x)  +  ■■■  -\-  E,(-i-)  =  0, 

where  R^,  R^,  ■■■  Rk  denote  rational  functions  of  x.  If  all  of 
these  functions  are  integral  rational  functions,  the  left 
member  of  (1)  is  an  integral  rational  function  of  .r,  and  we 


242  FRACTIONAL   RATIONAL   FUNCTIONS       [Anr.  145 

have  before  us  the  problem  discussed  in  Chapters  IV  and  V. 
Let  us  assume,  therefore,  that  at  least  one  of  the  functions 
i?j,  i^g,  •  ••  -R/fc  is  a  fraction.  If  one  or  several  of  these  terms 
are  fractions,  let  us  assume  that  each  one  of  them  is  written 
in  its  lowest  terms.  Let  us  assume  further  that  no  two  of 
these  fractions  have  the  same  denominator.  If  two  such 
terms  with  a  common  denominator  should  occur,  we  could 
unite  them  into  a  single  term. 

The  sum  of  the  rational  functions  R-^{x^,  B^^x'),  •••,  M^^r^ 
is  again  a  rational  function  which  may  be  obtained  by  reduc- 
ing R-^(x),  R^(x')^  •••,i?;t(^)  to  a  common  denominator  and 
afterward  adding  the  resulting  numerators.  The  sum  will 
not  be  in  its  lowest  terms  unless  we  use  for  this  purpose  the 
lowest  common  denominator,  that  is,  the  lowest  common  mul- 
tiple of  the  denominators  of  R^(x^,  R^(x)^  '••,RkQc).  But 
even  if  we  do  use  the  lowest  common  denominator,  we  can- 
not be  certain  that  the  sum  is  in  its  lowest  terms.  If  we 
have,  for  instance, 

1  2 

■^^^''^  =  (:,-3)2(:.-2)'   ^2^"^^  =  (.,-3)2(.r-5)' 

the  lowest  common  denominator  is  (.r  —  8)^(2;—  2) (2;—  5). 
The  sum,  R^{x^  +  R^(x^^  then  becomes 

^x-  9 ^_ Z(x-  3) 

(x  -  ^)\x  _  2)(a;  -  5)  -  (:c  -  '6)\x  -  2) (a;  -  5) 

and  is  not  in  its  lowest  terms. 

If,  however,  we  express  every  one  of  the  rational  fractions 
which  occurs  in  (1)  as  a  sum  of  simple  partial  fractions,  then 
unite  all  of  the  partial  fractions  which  have  the  same  denomi- 
nator into  a  single  one,  and  finally  add  these  ^jartial  fractio7is 
together,  using  as  a  common  denominator  the  loivest  common 
multiple  of  the  denominators  of  the  simple  partial  fractions,  we 
may  he  sure  that  the  sum  R{x^  obtained  in  this  way  is  in  its 
lowest  terms. 

Let  R(x)  be  the  sum  obtained  in  this  way.  We  may 
write 


Art.  U5]       FRACTIONAL   RATIONAL   EQUATIONS  243 

where  f(x)  and  g(x^  are  integral  rational  functions  which 
have  no  common  factor  involving-  x,  since  R(^x)  is  in  its 
lowest  terms.     The  equation  (1)  will  be  replaced  by 

(2)  i2(^)=/M  =  o, 

and  it  will  be  satisfied  by  all  of  the  roots  of  the  integral 
rational  equation 

(3)  fix)  =  0. 

Moreover,  no  other  value  of  x  will  satisfy  equation  (1). 

Therefore,  all  of  the  finite  roots  of  a  fractional  rational 
equation  may  he  obtained  by  solving  a  certain  integral  rational 
equation. 

The  process  of  deriving  (3)  from  (1)  is  commonly 
described  as  "  clearing  of  fractions.''^  But  unless  we  clear  of 
fractions  by  the  process  indicated  above,  first  resolving  each 
term  into  a  sum  of  simple  partial  fractions,  the  resulting 
integral  rational  equation  may  not  be  correct.  For,  unless 
the  sum 

B^ix)  +  B,{x)  +  ■■•  +  Rd-r)  =  Rix)  ='t^ 

is  in  its  lowest  terms,  we  cannot  conclude  that  all  of  the 
roots  of  (3)  will  also  be  roots  of  (1),  since  some  of  these 
roots  would  then  also  cause  the  denominator  g(x)  to  vanish. 
If  we  are  careful,  however,  while  clearing  of  fractions  not  to 
divide  by  any  integral  rational  expression  involving  x,  we 
may  be  sure  that  the  resulting  integral  rational  equation 
will  contain  all  of  the  roots  of  (1).  But  it  may  have  other 
roots  besides. 

We  may  formulate  some  of  these  results  more  briefly  by 
introducing  the  following  terms. 

A  second  equation,  derived  from  a  first  equation,  is  said 
to  be  equivalent  to  the  first,  if  it  lias  exactly  the  same   roots 


244  FRACTIONAL   RATIONAL   FUNCTIONS       [Art.  146 

as  the  first.  It  is  said  to  be  redundant  if  the  roots  of  the 
first  equation  are  included  among  its  roots,  and  if  it  has 
other  roots  besides.  The  second  equation  is  called  defective 
if  its  roots  do  not  include  all  of  the  roots  of  the  original 
equation. 

We  have  seen  that  there  always  exists  an  integral  rational 
equation  equivalent  to  a  given  fractional  rational  equation. 
But  we  have  also  seen  that  the  usual  process  of  finding  this 
equation  hy  the  method  of  '•'- clearing  of  fractions'"  is  not  always 
reliable.  If.,  however.,  in  clearing  of  fractions.,  we  carefully 
refrain  from  dividing  hy  an  integral  rational  function  of  x,  we 
may  he  sure  that  the  resulting  equation  loill  not  he  defective. 
We  can  decide.,  after  solving  this  equation.,  whether  it  is  redun- 
dant or  equivalent  hy  testing  each  of  its  roots  to  see  whether  all 
of  them  do  or  do  not  satisfy  the  original  fractional  equation. 

Special  care  is  necessary  to  avoid  dividing  by  a  factor  which  contains 
X.      Inexperienced  people  often    conclude  from    such    an   equation    as 

x(x  -  5)  =  0, 

that  X  =  5  is  the  only  solution.  But  x  =  0  is  also  a  solution  of  this 
equation. 

EXERCISE   LXVI 

Reduce  the  following  equations  to  integral  rational  equations. 
Discuss  the  question  of  equivalence,  and  solve. 

X  7  ^3,12 


X  +  60 

3 

x-5 

X  —  0 

27 

X 

=  13. 

8x 

6  = 

20 

x  +  2 

3x 

146.  Pressure  exerted  by  gases.  Let  us  suppose  that  the 
cylinder  CC  in  Fig.  59  contains  a  certain  volume,  say  Vq 
cubic  feet,  of  air  when  the  movable  piston  P  occupies  its 
highest  position,  and  let  us  compress  the  air  by  i)ushing  the 
piston  down  into  the  position  shown  in  the  figure.  As  we 
push  tlie  piston  down  farther  and  farther,  we  find  that  the 
resistance  of  the  inclosed  air  becomes  greater  and  greater. 


Art.  146]  PRESSURE   EXERTED  BY  GASES  245 

Thus  the  inclosed  air  exerts  a  force  tending  to  move  tlie 
piston  upward,  and  this  force  is  overcome  by  the  muscular 
force  which  we  exert   in   pushing   the   piston 


~'^''. — 1 


c 


down.  ^ 

In  Fig.  59  the  air  has  been  compressed  to 
about  one  third  of  its  original  volume.  If  we 
take  a  second  cylinder  of  the  same  height  but 
of  smaller  cross  section  and  compress  the  air 
in  it  to  one  third  of  its  original  volume,  we 
notice  that  a  smaller  force  will  suffice  for  this 
purpose  than  in  the  case  of  the  first  cylinder. 
In  fact,  measurement  of  these  forces  will  sliow  that  they  are  to 
each  other  as  the  cross  sections  of  the  two  cylinders.  Therefore, 
the  force  which  the  air  exerts  upon  a  unit  of  area  will  be  the 
same  in  both  cylinders.  The  force  which  any  gas  exerts  upon 
a  unit  area  of  a  containing  vessel  is  called  the  pressure  of  the 
gas,  and  is  usually  denoted  by  p.  The  total  force  which  a 
gas  exerts  upon  an  area  of  A  square  units  will  be 

(1)  F=Ap 

if  p  denotes  the  pressure  of  the  gas.     Consequently  we  have 

(2)  P  =  f. 

Since  a  unit  of  force  has  the  dimensions  ML/T^  (Art.  80) 
and  a  unit  of  area  has  the  dimensions  I?  (Art.  41),  a  unit  of 
pressure  according  to  (2)  has  the  dimensions 

(3)  M/LT^. 

When  gas  is  compressed  in  a  cylinder,  as  shown  in  Fig. 
59,  the  pressure  increases,  the  volume  decreases,  and  ordi- 
narily the  temperature  of  the  gas  rises.  If  the  cylinder  is 
made  of  metal,  we  may  wait  until  the  original  temperature, 
that  of  the  room  in  which  the  experiment  takes  place,  has 
reestablished  itself.  This  may  be  done  by  inclosing  a 
thermometer  in  G-  which  may  be  observed  from  the  outside 
through   a   glass  window.     We  are   thus  in    a    position  to 


246  FRACTIONAL   RATIONAL   FUNCTIONS      [Art.  146 

measure  the  various  pressures  which  the  gas  exerts  upon 
the  piston  at  the  various  moments  when  its  volume  has  dif- 
ferent values  while  its  temperature  remains  the  same.  Ex- 
periments of  this  general  character  were  first  performed  by 
Boyle  in  1661,  and  by  Mariotte  in  1676.  These  experi- 
ments lead  to  what  is  usually  known  as  Boyle's  law.  If  the 
volume  of  a  gas  is  changed  isothermallg,  that  is,  without  chang- 
ing its  temperature,  its  pressure  ivill  change  in  such  a  way  that 
the  product  of  pressure  and  volume  remains  constant. 

Thus  the  product  pv  can  depend  (for  a  given  mass  of  a 
given  kind  of  gas)  only  upon  the  temperature.  The  de- 
pendence of  this  product  upon  the  temperature  was  investi- 
gated by  Gay-Lussac  in  1802.     He  found  the  equation 

(4)  pv=RT,OTp  = , 

V 

where  T,  the  so-called  absolute  temperature  of  the  gas,  is 

(5)  ^=273  +  ^, 

6  being  the  temperature  expressed  in  Centigrade  degrees, 
and  where  i2  is  a  constant  which  depends  upon  the  amount 
of  gas  which  is  being  used  and  upon  the  chemical  constitu- 
tion of  the  gas. 

As  a  matter  of  fact,  equation  (4)  is  not  perfectly  exact. 
For  high  pressures  it  fails  to  agree  completely  with  the  ex- 
periments, and  the  following  equation,  due  to  Van  der 
Waals,  is  more  nearly  in  accordance  with  the  facts : 

RT      a 

V  —  0       v^ 

where  a  and  h  are  small  quantities  which  are  practically  un- 
noticeable  except  when  v  becomes  very  small.  Observe 
that  (6)  reduces  to  (4)  wlien  a  and  b  are  equal  to  zero. 
If  the  temperature  remains  constant  while  the  gas  is 
changing  its  volume,  eacli  of  the  equations  (4)  and  (6)  de- 
fines p  a,s  a.  fractional  rational  function  of  v. 


Art.  140]  TRESSURE   EXERTED   BY  GASES  247 

EXERCISE   LXVII 

1.  Use  a  unit  on  the  x-axis  to  represent  a  unit  of  volume  and  a  unit 
on  the  y-axis  to  represent  a  unit  of  pressure.  For  a  given  temperature 
represent  the  relation  expressed  by  the  combined  law  of  Boyle  and  Gay- 
Lussac  graphically.  Are  negative  values  of  p  and  v  admissible?  How 
will  the  curve  change  if  the  temperature  changes? 

2.  Investigate  the  form  of  the  curve  given  by  Van  der  Waal's  equa- 
tion. If  appropriate  units  are  used,  we  have  II  =  0.003G9,  a  =  0.00874, 
b  =  0.0023  when  the  gas  is  carbon  dioxide.  Trace  the  curves  which 
correspond  to  7'  =  250°,  300°,  350°. 


I 


CHAPTER   VII 

IRRATIONAL   FUNCTIONS 

147.  Existence  of  irrational  functions.  All  of  the  functions 
which  we  have  studied  so  far  were  rational  functions.  But 
there  exist  many  important  functions  which  are  not  rational. 
The  function  Va;  is  such  a  function.  However,  the  fact  that 
Va;  is  not  a  rational  function  of  x  requires  proof.  It  does 
not  follow  from  the  mere  presence  of  a  radical  sign. 

Let  a;  be  a  variable  capable  of  assuming  all  possible  real 
and  complex  values.  Every  number  x  has  two  square  roots 
each  of  which  is  determined  by  the  value  of  x  and  is  there- 
fore a  function  of  x.  Let  V^  denote  either  of  these  square 
roots.  We  observe  first  that  Va:  will  he  finite  whenever  x  is 
finite.  To  prove  this  statement,  let  x  assume  any  finite  com- 
plex value  whose  modulus  is  r  and  whose  amplitude  is  6. 
(See  Art.  30.)  Then  r  is  a  finite  positive  number.  Both 
square  roots  of  x  will  have  the  positive  finite  number  Vr  as 
their  common  modulus,  and  their  amplitudes  will  be  6/2  and 
^/2-h180°  respectively.  (Art.  120.)  In  other  words,  if  x 
is  finite,  both  square  roots  of  x  are  also  finite. 

We  shall  now  show  that  'Vx  is  not  a  rational  function  by 
the  method  of  reductio  ad  absurdum.  Suppose  it  were  a 
rational  function.     We  should  then  have,  for  all  values  of  a:. 


(1)  Vi  = 


/(^) 


where  /  (a;)  and  g(x~)  are  integral  rational  functions,  which 
we  may  assume  to  have  no  common  factor  involving  x. 
(Definition  of  a  rational  function  in  its  lowest  terms.  Arts. 
135  and  137.)  According  to  the  fundamental  theorem  of 
Algebra  (Art.   126),  there  exist  finite  values  of  x  (real  or 

248 


Am.  U7]     EXISTENCE   OF   IRKATIOXAL   FUNCTIONS        249 

imagiiKii'}')  wliicli  satisfy  the  equation  g(x^=0,  if  the  func- 
tion g(^x)  involves  x  at  all.  Such  values  of  x  would  cause  the 
right  member  of  (1)  to  become  infinite.  But  we  have  just 
seen  that  the  left  member  of  (1)  remains  finite  for  all  finite 
values  of  x.  Consequently,  equation  (1)  gives  rise  to  a  con- 
tradiction unless  (/{x^  does  not  involve  x  at  all,  that  is,  unless 
ff(x)  reduces  to  a  constant.  In  that  case,  the  quotient 
f  (x^lg{pc)  would  reduce  to  an  integral  rational  function  of 
a",  and  we  should  have,  for  all  vahies  of  x, 

(2)  V~v  =  a^-\-  a^x  +  a^x9-  +  •  ■  •  +  a^x"" 
or,  squaring  both  members, 

(3)  x  =  a^^^l  a^a-^x  +  (a^^  +  2  a^a^^x^  4-  •  •  •  +  «„V". 

But  an  identical  equation  of  form  (3)  can  hold  only  if  the 
coefficients  of  like  powers  of  x  in  tlie  two  members  are  equal 
to  each  otiier.  (See  Art.  126,  Theorem  F.)  Therefore  (3) 
can  be  true  for  all  values  of  x^  only  if 

0  =  Uq^,     1  =  2  a^a^,     0  =  a^"^  -{^  2  a^a^.,  etc. 

But  the  first  and  second  of  these  equations  contradict  each 
other.  Therefore  equation  (2)  is  impossible  and  Va;  is  not 
a  rational  function  of  x. 

We  could  prove,  in  similar  fashion,  that  Vx^  Va^,  •••  '\/ .r 
are  not  rational  functions  of  a:,  and  more  generally  that  the 
function  Vx""  is  not  a  rational  function  of  x  uidess  m  happens 
to  be  an  integral  multiple  of  n.  We  shall  speak  of  all  of 
these  functions  as  irrational  functions,  in  accordance  with  the 
following  definition. 

A  function  which  cannot  he  expressed  either  as  an  integral 
rational  fwiet ion  or  as  a  quotient  of  tivo  integral  rational  func- 
tions^ is  called  an  irrational  function. 

Tt  would  be  entirely  erroneous  to  conclude  that  the  jDresence  of  a 
radical  sign  in  the  expression  which  defines  a  function  is  either  necessary 
or  sufficient  to  make  the  function  irrational.  Thus  \/4  .r^  involves  a 
radical  sign,  but  this  function  is  equal  to  ±  2  x  and  is  therefore  rational. 


250  IRRATIONAL   FUNCTIONS  [Art.  148 

On  the  other  hand  the  trigonometric  functions,  sin  x,  cos  x,  etc.,  are  irra- 
tional although  they  involve  no  radical  signs.  In  applying  the  above 
definition  we  are  not  concerned  with  the  form  in  which  the  function 
happens  to  be  written.  The  real  question  is  whether  it  is  or  is  not 
possible  to  rewrite  it  in  such  a  way  as  to  make  it  assume  the  form  of  a 
rational  function. 

148.   The  function  y=\/x  and  its  principal  value.     If  a;  is 

a  given  number,  and  if  t/  is  another  number  sucli  that 

(1)  ?/"  =  X, 

y  is  said  to  be  an  wth  root  of  x  and  we  write 

(2)  y  =</x. 

We  have  seen  in  Art.  120  that  a  number  x  has  n  of  these  nth 
roots,  so  that  the  function  -s/x  may  be  regarded  as  having  n 
values  for  every  value  of  x.  It  is  an  n-valued  function.  If  x 
is  an  imaginary  number  none  of  tlie  n  values  of  Vx  are  real. 
If  X  is  real,  at  most  two  of  these  n  values  are  real.  (See  Art. 
120.) 

We  shall  henceforth  confine  our  attention  to  the  case  where 
both  of  the  variables,  x  and  ^,  in  (1)  are  real.  The  discus- 
sion of  this  case  naturally  breaks  up  into  two  sub-cases 
according  as  n  is  odd  or  even. 

Case  1  (ji  odd).  If  n  is  odd  we  may  give  to  x  any  real 
value,  positive  or  negative.  For  positive  values  of  a;,  equa- 
tion (1)  has  one  and  only  one  real  root,  and  this  one  is 
positive.  For  negative  values  of  x,  equation  (1)  again  has 
only  one  real  root,  but  this  time  the  real  root  is  negative. 
Consequently  we  shall  agree  from  now  on  to  use  the  symbol 
^x  in  the  following  more  precise  sense. 

If  n  is  an  odd  integer,  and  x  is  any  real  number^  the  sym- 
bol Vx  shall  stand  for  the  real  value  of  the  nth  root  of  x.  This 
real  value  of  ^\rx  (n  being  odd')  is  positive  or  negative  according 
as  x  itself  is  positive  or  negative. 

Case  2  (w  even).  If  n  is  even,  negative  values  of  x  are 
not  admissible  if  we  wish  to  discuss  only  real  functions  of  x. 


Art.  149]  THE   GRAPH    OF   //  =  Vx  261 

For  if  X  is  negative  and  n  is  even,  there  exists  no  real  wth 
root  of  X.  If  X  is  positive  and  n  is  even,  there  are  two  real 
/ith  roots  of  x^  one  positive  and  one  negative.  We  therefore 
make  the  following  agreement  so  as  to  avoid  all  ambiguity. 

If  n  is  an  even  integer,  the  function  Vx  is  defined  as  a  real 
function  of  x  only  if  x  is  positive.  Moreover,  x  being  positive, 
the  symbol  ^x  shall  be  used  to  stand  for  the  positive  nth  root 
of  X.      The  negative  nth  root  of  x  will  then  be  denoted  by  —  ^x. 

As  a  result  of  these  agreements,  the  function 

y  =  Vx 

is  noiv  defined  as  a  real  one-valued  function  for  all  real  positive 
or  negative  values  of  x  lohen  n  is  odd,  but  merely  for  all  positive 
values  of  x  ivhen  n  is  even. 

The  one- valued  function  obtained  in  this  way  shall  be 
called  the  principal  value  of  the  w-valued  function  defined  by 
equation  (1). 

149.  The  graph  oi  y  =  Vx.  In  constructing  the  graph  of 
such  a  function  we  should  remember  that  any  two  numbers, 
X  and  y,  which  satisfy  the  equation 

(1)  y  =  Vx, 
also  satisfy  the  equation 

(2)  y-  =  x. 

Now  the  graph  of  the  latter  equation  is  most  easily  con- 
structed by  choosing  the  values  of  y  lirst  and  then  computing 
the  corresponding  value  of  x,  rather  than  the       +y^,    p,    g 
other   way  around.     Moreover,  the  graph   of 

(2)  is  very  closely  related  to  the  graph  of 

(3)  y  =  x\ 

In  fact,  from  every  point  of  the  graph  of  (3)  ^'"-  *^' 

we  may  obtain  a  point  on  the  graph  of  (2)  by  interchang- 
ing its  x  with  its  y.  Now  two  points,  P  and  P'  (see  Fig. 
60),  whose  coordinates  are  so  related  that  the  abscissa  x'  of 
P'   is  equal  to  the  ordinate  y  of  P,  and  that  the  ordinate 


252  IRRATIONAL    FUNCTIONS  [Art.  149 

y'  of  P'  is  equal  to  the  abscissa  x  of  P,  are  symmetrically 
situated  with  respect  to  the  line  OB  which  bisects  the  angle 
between  the  positive  x-  and  y-axes.  This  relation  between 
P  and  P'  is  the  same  as  that  between  an  object  and  its  re- 
flected image,  if  OB  be  regarded  as  a  mirror.  We  shall 
therefore  speak  of  the  operation  which  replaces  the  point  P 
of  Fig.  60  by  P'  as  a  reflection. 

We  may  tlien  say,  that  the  graphs  of 

(4)  y  =  x^  and  of  //"  =  x 

are  reflections  of  each  other  with  respect  to  the  bisector  of  the 
angle  between  the  positive  coordinate  axes. 

The  two  functions  of  x  defined  by  (4*)  are  said  to  be  in- 
verse to  each  other,  and  the  relation  between  their  graphs, 
which  has  just  been  mentioned,  holds  for  any  pair  of  inverse 
functions. 

If  in  the  equation 

(1)  y  =  ^x, 

the  symbol  -y/x  stands  for  the  jirincipal  value  only  of  the  wth 
root  of  a;,  the  graph  may  be  07dy  a  part  of  the  graph  of  the 

equation 

^  y^  z=  X. 

Thus  the  graph  of 

y-  =  2: 

is  the  whole  parabola  of  Fig.  61,  which  has  the 
X-axis  as  its  axis  of  synnnetiy.  But  the  graph  of 
the  principal  value  of 

y  =  \^x 
is  merely  the  upper  portion  of  this  parabola,  that 
portion  whose  points  have  positive  ordiiiates. 


Fig.  CI 


EXERCISE   LXVIII 

Draw  graphs  of  the  following  functions,  interpreting  the  symbol   \/x 
to  mean  the  principal  value  in  each  case : 

1..   y  —  —  y/x.                       5.    y  =  vx.  9.  //  =x  +  Vx. 

2.  y  =  2Vx.                        6.    .y  =  —  Vx.  10.  7/  =  x  —  Vx. 

3.  y  =  \yfx.                        7.    ?/  =  2\/x.  11.  //  =  X  +  V—  X. 

4.  y  =  —  2Vx.                    8.   3/=—  iVx.  12.  ?/ =  x  —  V— x. 


Art.  1.-)0]  THE   FUNCTION   y  =  Vx"'  253 

150.  The  function  y  =  Vx"*.  As  a  result  of  the  agree- 
ments made  in  Art.  148  the  function  y/x  was  defined  as  a 
real  one-valued  function  of  the  real  variable  x  for  all  real 
values  of  x  when  n  is  odd,  and  for  all  positive  values  of  x 
when  n  is  even.     The  function 

(1)  ^^(Vxr 

wliere  7ti  is  a  positive  integer  will  then  also  be  defined  as  a 
real  one- valued  function  of  x,  for  all  real  values  of  x  when 
)i  is  odd,  but  only  for  all  positive  values  of  x  when  w  is  even. 
Let  us  raise  i/  to  the  nth.  power.     We  find 

^"  =  [(  Va?^)"]"  =  (a/'^)"'"  =  [(-v/ic)"]"'  =  a:"*, 
if  we  make  use  of  the  familiar  formulas 

(«">)"  =  (a")""  =  a'"", 

where  m  and  n  are  positive  integers.* 

Consequently,  the  one-valued  function  (1)  satisfies  the 
equation 

(2)  ^"  =  a:"*. 

But  this  equation,  which  is  of  the  nth  order  in  y,  has  n 
roots  (Art.  120)  each  of  which  is  called  an  wth  root  of  x"". 
A  consistent  use  of  the  symbols  would  lead  us  to  denote  any 
one  of  these  roots  by  Vx"'.  But  our  desire  to  have  this 
symbol  defined  unambiguously  leads  us  to  the  following 
more  specific  definition. 

The  symbol  ^/x""  shall  be  used  to  stayid  for  that  particular 
one  of  the  nth  roots  of  x""  ichich  is  equal  to  (\/:c)"',  tvhere  the 
symbol  Vx  is  defined  as  a  one-valued  function  of  x  by  the 
agreements  laid  down  in  Art.   149. 

Thus  the  principal  value  of  \'x"'  is  associated  with  the 
principal  value  of   Vx  in  such  a  way  that  we  shall  have 

(8)  V^={^xy. 

*  The  proof  of  these  formulas  may  he  reviewed  by  referring  to  Art.  157. 


254  IRRATIONAL   FUNCTIONS  [Art.  151 

The  choice  which  we  have  made  of  the  principal  value  of 
Vx^  may  also  be  expressed  as  follows. 

Whenever  there  exists  only  one  real  nth  root  of  x^  {x  itself 
being  real),  this  real  root  is  the  principal  value.  Whenever 
there  are  two  real  nth  roots  of  a;'",  one  will  he  positive  and  one 
will  be  negative.  In  that  case  the  positive  nth  root  of  x^  is  the 
principal  value. 

EXERCISE  LXIX 

Draw  graphs  of  the  following  functions : 

1.  y  =  yf^.  5.    J/  =  2  +  Vz3. 

2.  y  --  Vx^.  6.    ^/  =  2  -  yJ~^. 

3.  ij  =  vx2.  7.  ?/  =  -  2  +  v;^. 

4.    y  =  -  \/x^.  8.    ?/  =  -  2  -  Vx3. 

Making  use  of  the  definition  of  the  jirincipal  values  of  y/x  and  of 
yjx'^  as  given  in  Arts.  148  and  150,  prove  the  statements  in  Exs.  9-12. 
9.    ^x"'  is   defined  as  a  real  one-valued  function  of  x  for  all  real 
values  of  x,  if  n  and  m  are  both  odd.     This  function  is  positive  or  nega- 
tive according  .as  x  is  positive  or  negative. 

10.  \/x"»  is  defined  as  a  real  one- valued  function  of  x  for  all  real 
values  of  x,  if  n  is  odd  and  m  is  even.     It  is  positive  for  all  values  of  x. 

11.  Vx"'  is  defined  as  a  real  one-valued  function  of  x  for  all  real 
values  of  x,  if  n  and  m  are  both  even.     It  is  positive  for  all  values  of  x. 

12.  Vx"*  is  defined  as  a  real  one-valued  function  of  x  only  for  values 
of  x  which  are  positive  or  zero,  if  n  is  even  and  m  is  odd.  For  all  such 
values  of  x  the  function  is  positive. 

13.  For  what  values  of  x  will  the  function  V(2  x  —  7)^  be  defined  as 
a  real  one-valued  function  ? 


14.  For  what  values  of  x  will  the  function  V(7  -  2  x)^  be  defined  as 
a  real  one-valued  function  ? 

15.  Formulate  the  four  statements  which  correspond  to  Exs.  9,  10, 

11,  12  for  the  function  \/—  x'". 

151.   Properties  of   radicals.     The  following  formulas  are 
used  very  frequently  in  calculations  involving  radicals : 

(A)  -v/^  =  (-v^^)-  (C)        -^^='V^, 

(B)  VV^  =  "'-v/^,  (D)   V^Vx^=V^, 


Art.  151]  PROPERTIES   OF   RADICALS  255 

The  first  of  these  equations  has  already  been  proved.  It  is 
identical  with  equation  (3)  of  Art.  150. 

Let  us  prove  equation  (B).  The  right  member  of  (B)  is 
the  principal  value  of  the  mnth  root  of  x,  and  is.  therefore,  a 
positive  number  whenever  x  is  positive.  The  left  member 
of  (B)  is  also  an  wmtli  root  of  x.  For  it  is  easy  to  verify 
that  the  mnth  power  of  v  Va:  is  equal  to  a;.  If  a:  is  positive 
the  left  member  of  (B)  will  also  be  positive,  each  of  the 
radical  signs  being  used  to  designate  the  principal  value  of 
the  corresponding  root.  Thus  each  of  the  two  members  of 
(B)  is  a  positive  mni\i  root  of  the  positive  number  x,  con- 
sequently these  two  members  must  be  equal.  For,  a  positive 
number  x  has  one  and  only  one  positive  7nnt\\  root,  namely, 
the  principal  value  of  that  root.  (See  Art.  120.)  We  have 
now  proved  the  correctness  of  equation  (B)  for  all  positive 
values  of  x. 

If  X  is  negative,  both  members  of  (B)  will  be  imaginary 
unless  m  and  n  are  both  odd.  In  the  latter  case  the  prin- 
cipal values  of  both  members  of  (B)  will  be  negative.  Thus, 
both  members  of  (B)  are  mnth  roots  of  the  negative  number 
x^  and  both  of  them  are  negative.  Therefore,  they  are  equal. 
For,  a  negative  number  has  only  a  single  real  mnth  root  and 
this  is  negative.      (See  Art.  120.) 

The  remaining  case  when  a-  =  0  requires  no  discussion. 
Equation  (B)  is  obviously  true  \i  x=0. 

Thus  we  see  that  equation  (B)  is  actually  true  for  all  real 
values  of  x  in  all  cases  in  which  both  members  of  (B)  are 
defined  as  real  functions. 

To  prove  formulas  (C),  (D),  and  (E)  we  may  proceed  as 
in  proving  (B).  The  details  of  the  proof  are  left  as  an 
exercise  for  the  student. 

The  student  may  iw.ssibly  have  some  doubt  in  his  mind  as  to  the 
necessity  of  proving  these  equations  at  all.  The  following-  example  will 
help  to  clarify  the  situation.     Unless  we  specify  that  the  radical  signs 


256 


IRRATIONAL    FUNCTIONS 


[Art.  152 


are  to  be  used  for  the  principal  values  of  the  indicated  roots,  we  should 
have  the  alternatives 

V4  =  ±  2,         V9  =  ±  3,         Vm  =±Q 

and  we  could  not  affirm  that  the  equation 

ViV9  =V36 
is  necessarily  true  ;  it  might  just  as  well  be 

ViVO  =  -  VM. 

In  all  of  the  equations  (A)  .  .  .  (E)  both  members  are  easily  seen  to  be 
like  roots  of  one  and  the  same  number.  Since  we  are  discussing  only 
the  case  when  the  quantities  concerned  are  real,  it  is  evident  at  once 
that  the  two  members  of  the  equation  are  either  equal,  or  else  numeri- 
cally equal  and  opposite  in  sign.  The  essential  result  which  we  have 
obtained  is  this:  both  members  of  each  of  the  equations  (A)  .  .  .  (E) 
will  actually  be  equal  not  only  in  magnitude  but  also  in  sign,  whenever 
the  functions  concerned  have  real  values,  provided  that  every  radical  sign 
which  occurs  is  used  for  the  principal  value  of  the  corresponding  root. 


EXERCISE   LXX 

Simplify  the  following  expressions  : 


1. 

Vl8.                           4.    V- 
\/-27^-                    5.    </a^^ 

10000, 

7.    v/a'""&"P. 

2. 

8.  :/    (>'f'^'<^^' 

3. 
9. 

v'lOOOO.                    6.    V25 
v/24  +  V54  -  V6. 

.  as/jiopi 

15. 
16. 
17. 
18. 

19. 
20. 

10. 
11. 

12. 

^  />3  +  \  /,,/■>       \  5e2 

v^a  y/b'y/c. 
(3+V5)(2-V5). 

(9  -7Vi3)(5  -  evTa). 

13. 
14. 

Va^c  +  a^d. 

Vv'ti4. 

^b    ^h 

152-  The  square  root  of  a  rational  function.  When  we 
iiLtcmpt  to  extract  the  square  root  of  a  ratioual  function 
R{x).,  two  essentially  different  cases  may  present  themselves. 
The  square  root  may  again  be  a  rational  function,  or  else 
it  may  not.  In  the  former  case  we  may  say  that  i?(.r)  is  a 
perfect  square. 


Akt.  152]     SQUARE   ROOT   OF   RATIONAL   FUNCTION        257 

Thus  the  rational  function 
J.  (x-iy(Sx  +  5r 

^  ^  (x  -  3)4(x  -  7)6 

is  a  perfect  square.     One  of  its  square  roots  is  the  rational  function 

.0)  (X -])(:}  x  +  5)^ 

^'^  (X  -  3)^(x  -  7)3 

The  other  square  root  is  this  same  function  multiplied  by  —  1. 

This  example  shows  quite  clearly  why  the  square  roots 
of  (1)  are  rational  functions,  namely,  because  all  of  the  ex- 
ponents which  occur  in  (1)  are  even  numbers.  Now  an^ 
rational  function  may  be  written  in  what  we  have  called  its 
factored  form  (Art.  141),  and  it  will  obviously  be  a  perfect 
square  if  all  of  the  exponents  which  occur  in  it  are  even 
numbers.     We  may  express  this  result  as  follows  : 

I.  If,  in  the  factored  form  of  a  rational  function,  every  linear 
factor  of  its  numerator  and  of  its  denominator  appears  raised 
to  an  even  power.- the  function  is  a  perfect  square;  that  is,  both 
of  its  square  roots  will  be  rational  functions. 

Let  R{x)  be  a  rational  function.  Let  us  assume  that 
RQx^  has  been  written  in  its  factored  form,  and  that  not  all 
of  the  linear  factors  of  R(^x)  which  appear  in  this  factored 
form  are  raised  to  even  powers.  Then  each  of  the  square 
roots  of  i?(a;)  will  be  an  irrational  function.     For  let 


C'3)  i/=VR(x), 

be  one  of  these  square  roots,  so  that 
(4)  f=R(ix^- 

Equation  (4)  shows  that  y  cannot  be  a  rational  function. 
For  if  it  were,  i/^  in  its  factored  form  would  contain  only 
even  exponents,  contrary  to  the  assumption  that  7?(.r),  in  its 
factored  form,  contains  some  odd  exponents.  We  have 
obtained  the  following  theorem: 

IL  If,  in  the  factored  form  of  a  rational  function,  any  of 
the  factors  appear  raised  to  odd  powers,  both  of  the  square  roots 
of  the  function  will  be  irrational. 


258  IRRATIONAL   FUNCTIONS  [Art.  15.} 

Tlius,  the  function 

is  not  a  perfect  square  because  not  all  of  the  exponents  are  even.     The 
two  square  roots  of  R(x)  may  be  expressed  as  follows, 


(6)  v7fw  =  ±ii^iKM?Vj5l. 

This  example  illustrates  the  following  general  theorem. 

III.  If  a  rational  function  is  not  a  perfect  square,  its  square 
root  may  he  expressed  as  a  product,  one  of  whose  factors  is 
rational,  while  the  other  factor  is  a  square  root  of  a  rational 
function  with  simple  zeros  and  poles. 

The  first  of  these  factors  ma}%  of  course,  reduce  to  a  mere  constant, 
and  this  constant  may  be  equal  to  unity.  This  is  the  case  in  the  follow- 
ing example, 


X  ~1  \x  —i 


The  expression  for  the  square  root  of  a  rational  function 
may  be  simplified  still  more. 

In  the  expression  (6),  let  us  multiply  both  terms  of  the  fraction  which 
occurs  under  the  radical  sign  by  x  —  7.     We  find 


(x  -  ly 

where  we  now' have  an  integral  rational  function  with  distinct  linear 
factors  under  the  radical  sign.  The  corresponding  general  theorem  is 
as  follows : 

IV.  If  a  rational  function  is  not  a  perfect  square,  its  square 
root  may  he  expressed  as  a  product,  one  of  tvhose  factors  is 
rational,  while  the  other  is  a  square  root  of  an  integral  rational 
function  without  repeated  factors. 

153.  Functions  which  involve  the  square  root  of  a  rational 
function  and  no  other  irrationality.  Let  R(x)  he  a  rational 
function  which  is  not  a  perfect  square,  so  that  -\/ R(x)  is 


Art.  153]     SQUARE   ROOT   AS   ONLY  IRRATIONALITY       259 

irrational.     Consider  a  function  /  {x}  which  is  made  up  of 
a  sum  of  terms  of  the  form 


(1)  A,Cx},    A,ix)^E{x),    ^2(a.)(Vi2(a;))2, 

^3(x)(V;ft(^)3,  etc., 
where  ^o(^)'  -^i(-'')^  ^2(-*')'  ^^^•'  ^^®  rational  functions  of  x. 


Since        (V7^0r))2  =  7e(:r),  (Vi2(a;))3  =  i2(2;)Vi2(a:),  etc., 

each  of  these  terms  will  be  either  a  rational  function,  or  else 
a  product  of  a  rational  function  times  ^I{{x).  Conse- 
quently the  sum  of  all  of  these  terms  will  be  an  expression 
of  the  form 


(2)  rational  function  -f  another  rational  funetion  x  V^(2;}. 

If  R(x^  is  not  an  integral  rational  function  without  repeated 
factors,  we  may,  as  in  Art.  152,  express  ^R(x^  in  the  form 
of  a  rational  function  times  ^r{x)  where  r(x')  is  an  integral 
rational  function  without  repeated  factors,  so  that  Vr(2;)  is 
irrational. 

Thus,  the  sum  of  any  number  of  terms  of  the  form  (1) 
may  finally  be  written  in  the  form 


(3)  /(x)  =  A(x)  +  B(^x)Vr(ix% 

where  A{x')  and  B(^x')  are  rational  functions  of  x,  and  where 
r{x)  is  an  integral  rational  function  without  repeated  factors. 
Suppose  we  have  a  second  function  ,g{x^  of  the  same  kind 
as /(a;).     It  may  then  be  written  in  the  form 


(4)  g{x)  =  Cix)  +  I)(x)  Vr(2:), 

where  C(x^  and  -Z)(x)  are  rational  functions. 

The  quotient  of  two  such  functions  ma}^  again  be  reduced 
to  the  same  form.     To  prove  this  we  write 

A  +  B^/r     A  +  B^r   C-BVr 


(5) 


C+BVr      C+D^r   C -  D^r 

^AC-  BDr  +  (BC-  AD^^r 

c^-mr 


260  IRRATIONAL   FUNCTIONS  [Art.  15i 

where  C^  —  I>h'  will  not  be  identically  equal  to  zero.     For, 
if  it  were,  we  should  have 

_  (72 

contrary  to  our  assumption  that  Vr(a^)  is  irrational.     We 
may  rewrite  (5)  in  the  form 

where  L(x)  and  M(x^  are  rational  functions. 

Thus,  ever//  function  of  x,  which  depends  upon  the  square 
root  of  a  rational  function  in  the  way  indicated,  may  he  re- 
tvritten  in  the  form 


(6)  L  (a-)  +  M(x:)  Vr{x) 

where  L{x^  and  M(pc)  are  rational  functions  arid  where  r(a-) 
is  an  integral  rational  function  with  no  repeated  factors. 

We  may  speak  of  (6)  as  the  normal  form  of  such  a  func- 
tion. The  process  used  in  (5)  for  reducing  an  irrational 
function  of  this  kind  to  the  normal  form  is  frequently  known 
as  rationalizing  the  denominator. 

If  a  function  contains  more  than  one  independent  square 
root,  the  normal  form  may  be  obtained  by  rationalizing  first 
with  respect  to  one  of  the  square  roots,  then  with  respect  to 
the  second,  and  so  on. 

EXERCISE  LXXI 

Reduce  the  following  functions  to  the  normal  form  : 
..      x^-hxy/x  4     3a:-  4V.r2  -  9 


3. 


2Vx  3x  +  4v'x2-9 

.5  +  6  x Vx  c     a  +  ftVx^  —  m^  _ 
X  —  2Vx  a  —  hy/x"^  —  m'^ 

3+4  \\  -  x2  -       7  a;  +  5  +  ^'x^  -  x^  +  1 


,3  _  4  Vl  -  x2  3  x  +  2  +  4Vx8  -  x^  +  1' 


154.  Irrational  equations  of  the  simplest  type.  Let  f(x) 
be  an  irrational  function  of  x.  Tiiis  function  may  assume 
the    value  zero   for   one  or  several   particular  values  of   x. 


Akt.  1:)4]        simplest   IRRATIONAL   EQUATIONS  261 

These  values  of  .r,  if  they  exist,  are  called  the  zeros  of  f(j-), 
or  the  roots  of  the  irrational  equation 

(1)  /(•r)  =  0. 

Let  us  confine  our  attention  to  the  case  where  the  function 
/■(.r)  contains  as  its  only  irrationality  the  square  root  of  a 
single  rational  function.  We  have  just  shown  (in  Art.  153) 
that  we  may  then  write /(i-)  in  the  form 


/(.r)  =  i(r)  +  i>f(.r)Vr(^.i') 

wheic  L(x)  and  3I(x)  are  rational  functions,  and  where 
r(a-)  is  an  integral  rational  function  with  no  repeated  factor. 
Consequently  equation  (1)  will  assume  the  form 


(2)  L(^x)+  M(x)Vr{x)  =  0 
or 

(3)  L(x)=-  71if(x)Vr(a-). 

If  i-  is  a  root  of  (3),  it  will  also  satisfy  the  equation 

(4)  [X(.0]^  =  [i»f(.r)]V(.r), 

obtained  from  (3)  by  sc^uaring  both  members,  and  (4)  is  a 
rational  equation  which  may  be  solved  by  tlie  methods  of 
Art.  145.  If  we  remember  what  those  metliods  were,  we 
shall  recognize  the  truth  of  the  following  statement : 

The  problem  of  solving  an  irrational  equation  of  form  (2) 
77iai/  he  reduced  to  the  problem  of  solving  a  certain  integral 
rational  equation  in  .r. 

But  Avhile  it  is  certain  that  all  solutions  of  (2)  will  also  be 
solutions  of  tlie  rational  equation  (4),  it  is  not  at  all  certain 
that  every  solution  of  (4)  will  be  a  solution  of  (2). 

//■  thr  si/i)i/iol  \/r(.r)  stands  for  the  principal  value  <f  the 
square  root  <f  r{.r},  ctpiafians  (2)  and  (4)  will  in  general  not 
be  equivalent. 

In  fact,  not  only  tlie  roots  of  (2).  but  all  of  llie  roots  of 
the  (■(|nati(.)n 


262  IRRATIONAL   FUNCTIONS  [Art.  155 

(5)  X(.r)  -  If (.r)  V/<^  =  0, 

as  well,  will  satisfy  equation  (4).  Moreover,  since  from  (4) 
we  may  conclude 

L(^x)  =  ±  M(x}-Vr{x), 

it  is  clear  that  the  roots  of  (4)  will  be  distributed  among  the 
two  equations  (2)  and  (5).  It  may  happen  that  one  of  the 
equations  (2)  or  (5)  has  no  roots  at  all. 

These  remarks  justify  the  following  conclusion  :  The  solu- 
tion of  an  irrational  equation  which  involves  a  square  root  of  a 
rational  function  of  the  unknown  quantity/  and  no  other  irra- 
tionality may  he  reduced  to  the  solution  of  an  integral  rational 
equation.  But  in  general  the  latter  equation  will  not  he  equiv- 
alent to  the  origirial  equation^  and  each  of  its  roots  should  he 
tested  before  announcing  it  as  a  root  of  the  original  irrational 
equation. 

Similar  remarks  may  be  made  about  irrational  equations 
of  a  more  complicated  kind,  but  it  would  be  unprofitable  to 
discuss  such  cases  here. 

EXERCISE    LXXII 

Solve  the  following  equations  with  the  understanding  that  the  symbol 
Va  stands  for  the  positive  square  root  of  a. 

1.    ^'20^  -V^9  =  3.  3.    5x  =  39  +  2V^. 


2.     X  +V4:  +  x^  - 


10 


4.   X  +  Vx  —  4  =  1. 


V4  +  a:2  5.    -Vx  +  y/x  -9  =1. 

155.  The  general  algebraic  function.  All  of  the  functions, 
rational  as  well  as  irrational,  which  we  have  considered  so 
far,  belong  to  the  class  of  algebraic  functions,  which  may  be 
defined  as  follows  : 

Let  -4(a-),  B(x).,  •••,  i(a;),  M(x^  he  rational  functions  of  x. 
The  equation 

(1)         A^xyy"  4-  ^(a;)2/"-i  -}-•••+  L(^x)y  +  M(x)  =  0 


Art.  155]     THE   GENERAL    ALCIEBRATC    FUNCTION  263 

determines  y  as  an  n-valued  function  of  x,  since  according  to 
the  funda^nental  theorem  of  Algebra^  for  any  given  value  of  x 
the  equation  (1)  will  have  n  roots. 

Any  function  of  x  which,  when  substituted  for  y,  will 
satisfy  an  equation  of  the  form  (1  ),  witli  rational  functions 
as  coefficients,  is  called  an  algebraic  function.  All  other 
functions  are  called  transcendental. 

Any  function,  which  is  built  up  from  a  finite  number  of 
rational  functions  by  means  of  operations  involving  a  finite 
number  of  radical  signs,  is  an  algebraic  function  in  the  sense 
of  the  above  definition. 


y  =  X  +  vx-  +  y/x 

is  an  algebraic  f auction.     For  we  can  easily  find  an  equation  of  form  (1) 
which  this  function  will  satisfy.     We  have 

(//  —  x)^  =  x-  +  Vx 
and  therefore 

This  eqviation  when  expanded  is  of  the  form  (1),  its  degree  being 
n  =  6. 

But  not  every  algebraic  function  can  be  expressed  in 
terms  of  a  finite  number  of  rational  operations  and  extrac- 
tion of  roots.  We  know,  for  instance,  that  the  general 
equation  of  the  fifth  degree  cannot  be  solved  by  means  of 
such  operations.  (See  Art.  125.)  Consequently,  the  func- 
tions defined  by  means  of  an  equation  of  the  fifth  degree  in 
y,  with  rational  functions  of  x  as  coefficients,  cannot  always 
be  expressed  by  means  of  a  finite  number  of  radical  signs, 
although  these  functions  are  algebraic  according  to  our  defi- 
nition. 

For  this  reason,  algebraic  functions  are  frequently  divided 
into  two  classes.  Those  which  can  be  expressed  in  terms 
of  a  finite  number  of  radical  signs  are  called  explicit  alge- 
braic functions,  to  distinguish  them  from  the  more  general 
algebraic  functions  which  cannot  be  expressed  in  this  way. 

The  general  theory  of  algebraic  functions  is  a  very  impor- 


264  IRRATIONAL   FUNCTIONS  [Art.  155 

tant  and  interesting  field  of  study.  The  student  will  have 
the  opportunity  of  becoming  acquainted  with  some  of  the 
simpler  cases  of  algebraic  functions  in  his  course  in  analytic 
geometry,  and  he  is  already  in  possession  of  all  of  the  pre- 
liminaries that  will  enable  him  to  pursue  this  study  to 
advantage.  The  general  theory,  however,  is  a  far  more 
difficult  subject,  and  its  discussion  must  be  postponed  to  a 
much  later  point  in  the  student's  mathematical  career.  This 
general  theory  is  essentially  a  product  of  the  nineteenth 
century  and  is  due  principally  to  Cauchy  (1789-1857), 
Abel  (1802-1829),  Jacobi  (1804-1851),  Riemann  (1826- 
186G),  Weierstrass  (1815-1897). 


CHAPTER   VTII 

FRACTIONAL  AND  NEGATIVE  EXPONENTS 

THE  GENERAL  PO"WER  FUNCTION 

THE  EXPONENTIAL  FUNCTION  AND  LOGARITHMS 

156.  Fractional,  negative,  and  vanishing  exponents.  Tlie 
«tudejit  knoAvs,  from  las  first  course  in  Algebra,  that  the 
function  ^x^  is  frequently  written  with  a  different  notation, 
namely, 

(1)  V^  =  :f-'S 

and  the  symbols  7^  and  a;""  are  often  interpreted  in  accord- 
ance with  the  equations 

(2)  a;0  =  1,     x-^  =  \' 

The  reasons,  which  have  led  to  the  adoption  of  these  no- 
tations are  of  so  much  interest  that  we  shall  treat  the  matter 
as  thouo-h  we  were  now  meetino"  it  for  the  first  time. 

157.  The  index  laws.  If  m  is  a  positive  integer^  the  symbol 
.?■'"  is  used  to  de)iote  a  product  of  m  factors,  each  of  which  is 
equal  to  X.  This  definition  of  the  symbt)l  ./'"  may  be  ex- 
pressed as  follows : 

(1)  x^  =  x-x-X"-x  (a  product  of  m  c(|ual  factors,  m 
being  a  positive  integer).  From  this  dcliiiilion  tlic  follow- 
ing theorems  follow  at  once  : 

I.  X'"  •  .r'^  =  .r'""^"  (m  and  n  being  positive  integers). 

II.  (./•'")"  =  .^•""'  (w  and  n  being  positive  integers). 

III.  ^=x"'~"    (m    and    n    being    positive    integers,   and 

X" 

m  >  71 ). 

Formulas  I,  II,  III  are  usually  known  as  the  index  laws. 
The    definition    (1)  of  x""  is  entirely  devoid  of  meaning 

265 


266  THE   GEXERAL   POWER   FUNCTION         [Akt.  157 

unless  m  is  a  positive  integer.  For,  in  this  definition,  m 
stands  for  the  number  of  factors  in  a  certain  product,  and  it 
is  absurd  to  speak  of  a  product  composed  of  a  fractional  num- 
ber of  factors,  a  negative  number  of  factors,  or  no  factor 
at  all.  Consequently  the  definition  (1)  cannot  be  used  in 
any  case  in  which  m  is  not  a  positive  integer. 

Thus,  if  we  wish  to  give  a  meaning  to  the  symbol  a;'"  when 
m  is  not  a  positive  integer,  we  must  seek  a  new  delinition 
essentially  different  from  (1).  Now,  as  a  mere  matter  of 
logic,  rue  have  the  right  to  define  the  symbols  of  algebra  i)t  any 
way  we  may  desire.  Thus  we  might,  for  instance,  define 
the  symbol  x'^  in  such  a  way  as  to  make  it  stand  for  x/2,  for 
2/x,  for  Va;,  or  for  anything  else  we  please,  and  nobody 
would  have  a  right  to  object  to  any  of  these  definitions  on 
the  score  of  logic.  But  one  might  very  properly  object  to 
some  of  these  definitions  on  account  of  their  inconvenience. 

In  order  to  obtain  a  definition  for  x"^  which  shall  not  only 
be  logically  admissible,  but  which  shall  also  be  convenient 
and  useful,  we  argue  as  follows.  Nothing  compels  us  to 
introduce  such  a  symbol  as  a;^  at  all.  The  very  fact  that 
this  symbol  resembles  the  familiar  symbols  a-^,  x^,  x^,  etc.,  so 
much,  makes  it  inconvenient  to  introduce  it  at  all  unless  it 
can  be  done  in  such  a  way  as  to  enable  us  to  perform  calcu- 
lations with  this  new  symbol  according  to  the  same  rules 
(the  index  laws)  which  hold  for  x^,  x^,  a:^  etc.  Thus,  our 
desire  for  a  convenient  definition  leads  to  the  following  ques- 
tion :  Is  it  possible  to  define  the  symbol  x^  in  such  a  way  that 
the  index  laws  /,  /i,  and  III  may  he  used  m  all  of  the  calcula- 
tions in  which  this  new  symbol  occurs? 

This  question  is  easy  to  answer.     If  it  is  possible  to  define 
x^  in  such  a  way,  we  may  apply  index  law  II  to  it,  for  the 
purpose  of  computing  its  square,  giving 
(a;2-^2  _  ^2  _  ^^ 

Therefore,  the  symbol  x'^  must  be  taken  to  mean  one  of  the 
two  square  roots  of  x.*    If  x  is  positive,  one  of  its  two  square 

*  Since  its  square  is  equal  to  x. 


Ain.  157]  THE    INDEX    LAWS  2G7 

roots  is  positive  and  the  otlier  is  negative.  We  therefore 
define  x^  as  the  positive  square  root  of  x,  whenever  a:  is  a 
positive  number.  If  x  is  negative,  the  two  square  roots 
of  X  are  both  imaginary  and  either  one  of  them  may  be 

identified  with  the  symbol  x-. 

1 

In  the  same  way  we  are  led  to  define  a"  bi/  means  of  the 

equation 

1  _ 

(2)  x^  =  ^/x, 

the  principal  value  of  the  nth  root  of  x  being  meant  when  x  is 
real. 

(See  Art.  148  for  definition  of  principal  value.) 
According  to  the  second  index  law,  we  shall  have 

1  m 

(^xy  =  (x"y'  =  x\ 

m 

thus  leading  to  a  definition  for  x'\  namely., 

m  _  

(3)  x"=(Vxy=Vx'\ 

where  the  second  and  tliird  members  are  equal  on  account 
of  (3),  Art.  150. 

We  observe  that  the  third  index  law 

(4)  —  =  a;--" 
^  x^ 

was  subject  to  the  restriction  m  >  n.  If  ni  is  less  than  n  or 
equal  to  w,  the  exponent  in  the  right  member  becomes  nega- 
tive or  zero.  If  we  wish  to  introduce  negative  or  vanishing 
exponents  in  such  a  way  as  to  preserve  the  validity  of  the 
index  laws,  we  must  therefore  do  so  in  accordance  with  what 
equation  (4)  tells  us.  But  if  we  put  w  =  w  in  (4)  we  find 
the  following  definition  of  .r^  : 

(5)  ^  =  ?-  =  l. 


268  THE    GENERAL   POWER   FUNCTION         [Aim.  1.> 

If  m  is  less  than   /«,  let  it  be  equal  to  n—k.     If   we   put 
m  =  n  —  k  in  (4)  we  find 


X 
X 


-k 

^n—k — n  /y» — k 


But  the  left  member  of  this  equation  is  actually  equal   to 
l/a:^  thus  leading  us  to  write 

(6)  x-^  =  \ 

x" 

as  the  definition  of  x~^.     Combining  (3)  and  (6),  we  have 

-™       1  1 


JU  —         ,,„ 


x^ 

We  have  now  defined  the  s3'mbol  x^  in  all  cases  in  which 
r  is  either  a  positive  or  negative  integer  or  zero,  or  a  positive 
or  negative  rational  fraction.  We  know  that  no  other  defi- 
nitions than  those  actually  adopted  were  possible,  if  the  in- 
dex laws  were  to  be  satisfied.  But  we  are  not  yet  absolutely 
certain  that  all  of  the  index  laws  will  actually  be  satisfied  by 
these  symbols  in  all  cases.  For  we  have  only  used  one  of 
these  laws  for  the  purpose  of  guiding  us  toward  the  appro- 
priate definition  in  each  case.  It  is  possible,  however,  to 
verify,  by  actual  test,  that  the  powers  with  fractional,  7iegative, 
and  vanishing  exponents  defined  in  this  wa//  actually  obey  all 
of  the  index  laws.  We  may  therefore  operate  with  them  ac- 
cording to  the  same  rules  which  hold  for  positive  integral  ex- 
ponents. It  is  this  fact  which  makes  tliese  definitions,  not 
merely  logically  admissible,  but  extremely  convenient  and 
useful. 

158.  The  principle  of  permanence.  The  j)oint  of  view 
which  guided  us  in  fornndating  these  definitions  is  known 
as  the  priyiciple  of  the  permanence  of  theforvnal  latvs  of  algebra. 
This  same  principle  aided  us  in  Chapter  I,  when  we  were 
engaged  in  enlarging  the  number  system  of  algebra,  ])y  add- 
ing to  the  system  of  all  positive  integers  the  negative,  frac- 
tional, irrational,  and  complex  numbers.      The  principle  of 


Art.  158]         THE    PRINCIPLE   OF    PERMANENCE  269 

permanence  is  not  a  principle  of  logic.  It  is  a  heuristic 
principle  which  leads  us  to  new  definitions  and  notions, 
enabling  us  to  obtain  results  which  are,  not  merely  logically 
admissible,  but  simple  and  convenient.  Its  main  object  is 
to  make  a  few  formal  laws  (formulas)  suffice  where  other- 
wise there  would  be  many.  It  accomplishes  this  purpose 
by  making  a  formula  which  is  already  in  existence  do  more 
work  than  was  originally  intended  for  it,  thus  preventing 
the  introduction  of  a  new  formula. 

Thus,  in  the  present  instance,  the  index  hiws  were  originally  intended 
to  apply  only  to  powers  with  positive  integral  exponents.  The  principle 
of  permanence  has  led  us  to  adopt  such  definitions  as  to  make  the  index 
laws  accomplish  very  much  raoi-e.  They  now  apply  to  powers  with 
negative  and  fractional  exponents  as  well. 

EXERCISE  LXXIIi 
Find  the  value  of  the  following  numbers  : 

1.    8^         2.    3-^  3.    16"l        4.    (VO^-        5-    25"^.        6.    (fl)"^: 

Obtain  equal  expressions  free  from  negative  and  fractional  exponents : 
7.   3a~\  8.    Sx--b-^c^.  9.    (a-hc)\  10.    a'ph'^c'l- 

Perform  the  operations  indicated  and  simplify  : 

11.  (a^y*.  13.      "_)   •  15.    ^""^     ^ 

12.  (ahK^^y.  14.    (^\\  16.     ^f^" 


y  Va 


Perform  the  following  multiplications: 


21.  (a^+b^-){a'  -b-^). 

22.  (a^  +  //^)(a-  +  b^). 
b')(J  -  aW  +  6*). 


29.  a  —  b  by  a^  +  b'-. 

1         1 

30.  a  —  b  hy  a-  —  b^. 

31.    a-b  by  a^  +  aV  +  OK  32.    a  +  b  by  J  -  aM  +b^ . 


3      5 

17.   aia^. 

m    p 
19.      «"«?. 

18.    a'h-^ah'c. 

_m   p 

20.    a   'HiQ. 

23.    {a^-b^)(J 

+ 

a^b^  +  b'^).         24.    \ 

Divide : 

m              p 

25.    o»  by  as. 

m              p 

27.    a   «  by  ««• 

m               _p 

26.    a"  by  a  «. 

_m               _p 

28.    a   "  by  a   i. 

270  THE   GENERAL   POWER   FUNCTION         [Art.  159 

159.  The  case  of  an  irrational  exponent.  If  the  exponent 
k  is  an  irrational  number,  the  symbol  x''  is  as  yet  undefined. 
To  define  x'^  in  this  case  also,  we  proceed  as  follows. 

We  know  that  any  irrational  number  k  can  be  expressed 
with  any  desired  degree  of  approximation  by  a  decimal. 
(See  Art.  77.)  Let  k^  be  a  number  written  in  the  decimal 
notation,  with  n  figures  to  the  right  of  the  decimal  point, 
such  that  kn  is  less  than  k  but  differs  from  k  by  less  than  one 
unit  of  the  nt\\  decimal  place.  Let  kj  be  the  number  ob- 
tained from  kn  by  increasing  the  digit  in  the  nth.  decimal 
place  by  one  unit.     Then 

As  n  (the  number  of  decimal  places)  grows  beyond  bound, 
the  rational  numbers  k^  and  kj  both  approach  k  as  a  limit. 
It  can  be  shown  that  the  numbers 

/j-JCl  /yiA2  /y"3         ,   , 

/y»n'l  /yt^^'l  ry^.i 

JU         *      •</        ^      •c-         ^    "     * 

will  also  approach  a  common  limit.     We  define  xf'  to  he  this  limit. 

It  can  further  be  shown  (here  again  the  monotonic  laws 
turn  out  to  be  important),  that  the  index  laivs  will  continue 
to  hold  even  if  some  or  all  of  the  exponents  involved  are  irratiotial. 

It  is  not  difficult  to  prove  that  the  function  x''  is  a  tran- 
scendental function  (see  Art.  155)  if  k  is  an  irrational 
number.  This  fact  indicates  how  delicate  the  distinction 
between  algebraic  and  transcendental  functions  may  some- 
times be.  All  of  the  functions  x^\  x''\  •  •  •  ,  x^  ■  •  •  indicated 
in  the  above  argument  are  algebraic  and  the  exponent  k^  of 
^u  (liffers  from  k  by  less  than  one  unit  of  the  nth.  decimal 
place.     Nevertheless  a/'»  is  algebraic  and  x^  is  transcendental. 

We  know  that  V2  is  irrational.  The  numbers  denoted  above  by 
ki,  ki,  etc.,  will,  in  this  instance,  be  as  follows  : 

/.     _   1    4  _   14      7-    _  1    41    _   141      7.     _  1   414  _    1414    etc 

The  functions 

14         14  1  J.4 14 

a;i^  a;'*^,  a;i»«<^,  etc., 
are  all  algebraic,  but  the  function  x^  is  transcendental. 


Art.  160]  THE    POWER    FUNCTION  271 

160.   The    power    function.      We    have    now   defined   the 
function 

(1)  ?/=a:*= 

for  all  cases  in  which  the  exponent  ^  is  a  real  number.  This 
function,  which  we  shall  call  a  power  function,  reduces  to 
some  very  familiar  functions  if  we  give  special  values  to  the 
exponent  k.  Thus  for  A;  =  1,  we  find  the  simple  linear  func- 
tion ^  =  a;  whose  graph  is  a  straight  line  (see  Art.  51);  for 
^  =  2  we  find  the  simple  quadratic  function  y  =  x^  whose 
graph  is  a  parabola  (see  Art.  65).  Whenever  Ar  is  a  positive 
integer  x^  is  an  integral  rational  function  of  x  (see  Art.  82). 
If  ^  =  0,  x^  z=x^  =  1,  so  that  the  graph  of  (1)  in  this  case  is  a 
line  parallel  to  the  a;-axis  at  a  distance  of  one  unit  above 
it.     If  A;  =  —  1,  (1)  becomes 

_i     1 

y=X   1  =  -, 
X 

whose  graph  (an  equilateral  hyperbola)  we  studied  in  Art. 
140.  Whenever  A;  is  a  negative  integer,  y  =  x^  is  a  rational 
function  which  has  a;  =  0  as  a  pole  (see  Art.  139),  and  there- 
fore becomes  infinite  as  x  approaches  zero. 

If  ^  is  a  positive  rational  number,  so  that  Iz  =  mjn  where  m 
and  n  are  positive  integers,  we  have 


y 


=  2:*=  =  .r  "  =  Va 


according  to  Art.  157,  and  we  have  studied  such  functions 
and  their  graphs  in  Art.  150.  From  what  was  said  at  the 
end  of  Art.  159  it  is  clear  that  the  graph  of  the  function 
y  =  a;^  if  yfc  is  a  positive  irrational  number,  may  be  regarded 
as  approximately  the  same  as  that  of  a  function  of  the  form 
^m/n  ^yi^i-^  r^  rational  exponent,  provided  that  the  exponent 
m/n  is  a  sufficiently  close  approximation  to  k. 

Thus,  tlie  graph    of  //  =  x^-  is  approximately   the  same  as  tliat  of 

141  

y  =z  x^^^,  since  V2  is  approximately  equal  to  |  J^. 


272  THE   GENERAL   POWER   FUNCTION         [Art.  160 

If  ^  is  a  negative  rational  number,  so  that 

n 
where  m  and  n  are  positive  integers,  we  have 

m  "j 

y  =  x'^  =  x'n  =     ' ,  (See  Art.  157) 

a  function  which  becomes  infinite  for  x  =  0.  These  func- 
tions may  also  be  regarded  as  approximate  expressions  for 
the  case  where  Z;  is  a  negative  irrational  number. 

Thus  the  function  x-^'^  is  approximately  equal  to  x  ^^^,  or 
1  1 


We  shall  now  generalize  our  definition  of  power  function 
slightly,  as  follows : 

Any  function  of  the  form  ax^^  where  a  atid  k  are  constants, 
shall  be  called  a  poiver  function. 

Power  functions  have  an  important  property,  which  is 
common  to  all  of  them,  and  which  we  shall  now  deduce. 

Let  Xj,  OTg,  a^3,  •••  be  several  values  of  x  and  let  «/j,  y^-,  yy>  ••• 
be  the  corresponding  values  of  y.     Then  we  shall  have 

Vx  ^^  ax-^  ,     7/2  ^  ax2  ,     y^  =  ax^  •••, 
whence 

(2)  ^2  =  -V  =  (-hy^      !h  ^  ffsY,  etc. 

If   the   values   x^,   x^-,   3*3,  •••  form    a  geometric  progression 
whose  ratio  is  r,  we  hiive 


=  r 


and  therefore,  according  to  (2). 
^  =  ^=  ...^rK 


AuT.  160]  TUE   POWER   FUNCTION  273 

In  other  words,  ?/j,  i/^^  3/3,  •••  will  also  form  a  geometric 
progression,  whose  constant  ratio  r*  is,  of  course,  in  general 
different  from  r. 

We  obtain  the  following  theorem  which  expresses  the 
essential  property  common  to  all  power  functions : 

Let  y  he  a  pou'er  function  of  a-,  and  let  us  take  any  number 
of  values  of  x  which  form  a  geometric  progression.  The  cor- 
responding values  of  y  ivill  then  also  form  a  geometric  pro- 
gression. 

This  may  also  be  expressed  as  follows  :  In  a  power  func- 
tion., to  values  of  the  independent  variable  which  form  a 
geometric  progression,  there  correspond  values  of  the  function 
which  likewise  form  a  geometric  progrei<sion. 

This  theorem  enables  us  to  recognize  a  power  function  when  a  large 
number  of  pairs  of  values  of  x  and  y  are  given  numerically,  as  a  result 
of  experiment  and  measurement,  and  we  may  then  actually  find  a 
formula  for  the  relation  between  x  and  y,  expressing  y  as  a  power 
function  of  x. 

There  is  an  important  distinction  between  power  functions 
according  as  their  exponents  are  positive  or  negative,  a  dis- 
tinction which  has  already  appeared  implicitly  in  the  dis- 
cussion at  the  beginning  of  this  article. 

A  potver  function  y  =  ax''  for  which  the  exponent  k  is  positive 
asnimes  the  value  zero  tvhen  x  becomes  equal  to  zero.  But  a 
power  function  whose  exponent  is  negative  becomes  infinite  when 
X  approaches  zero. 

Therefore,  the  graph  of  a  power  function  with  a  positive 
exponent  passes  thi-ough  the  origin  while  the  graph  of  a 
power  function  with  a  negative  exponent  does  not  pass 
through  the  origin. 

The  two  functions  >j  =  x^,  and  y  =  ar-i  =  -  will   illustrate  this  state- 

X 

ment. 


274  THE    GENERAL   POWER   FUNCTION     [Akts.  161,  162 

161.  The  exponential  function.  Up  to  the  present  moment 
we  have  always  thought  of  the  exponent  in  the  equation 

(1)  y  =  x>' 

as  a  constant,  and  the  base  as  variable.  We  now  propose  to 
think  of  the  base  as  a  constant  and  the  exponent  as  variable; 
that  is,  we  propose  to  study  the  expression  (1)  as  a  function 
of  the  exponent.  We  shall  indicate  tliis  new  point  of  view 
by  a  change  of  notation,  writing  a  for  the  fixed  base,  and  x 
for  the  variable  exponent,  so  that  (1)  becomes 

(2)  y  =  a-. 

After  the  base  a  has  been  chosen,  the  value  of  a^  will 
depend  only  upon  x,  so  that  a""  is  a  function  of  x.  It  is 
called  an  exponential  function. 

We  shall  eo7isider  only  the  exponential  functions  whose  bases 
are  positive  numbers  different  from  unity.  The  reasons  for 
these  restrictions  are  fairly  obvious.  If  the  base  a  were 
negative,  a^  would  not  be  real  if  x  were  equal  to  1/2,  1/4,  1/8, 
or  any  fraction  with  an  even  denominator.  If  the  base  were 
equal  to  unity,  a^  would  be  equal  to  1^  and  this  would  be 
equal  to  1  for  all  values  of  x. 

We  agree.,  moreover.,  when  a  is  positive  and  when  the  exponent 
x  is  a  fraction  m/n,  that  a^  shall  always  be  defined  as  the  prin- 
cijyal  value  of 

a''  =  Va™, 

quite  in  accordance  with  what  ivas  said  in  Art.  157. 

In  most  applications,  the  base  a  is,  moreover,  taken  to  be 
greater  than  unity.     However,  this  is  not  at  all  essential. 

162.  Graphs  of  exponential  functions.  The  method  of 
constructing  the  graph  of  an  exponential  function  needs  no 
detailed  explanation.  It  is  precisely  the  same  as  for  all  of 
the  other  functions  which  we  have  studied  so  far. 


Art.  16:]] 


THE   EXPONENTIAL   FUNCTION 


275 


EXERCISE   LXXIV 

1.  Construct  the  graph  of  l?^. 

Solution.  We  compute  the  table  of  values  on  the  right- 
hand  margin  of  this  page,  by  putting 
1/  =  2"  and  computing  the  values  of  // 
which  correspond  to  the  values  x  =  —  4j 
-  3,  -  2,  -  1,  0,  +  1,  +  2,  +  :},  +  4. 
We  plot  the  corresponding  points  in 
Fig.  62  and  connect  them  by  a  smooth 
curve.  This  curve  is  the  required 
graph. 

Construct  the  graphs  of  the  following  functions  : 

2.  3"^.  5.    (V2)'^. 

3.  4*.  6.   2-'. 


Fig.  02. 


X 

-4 

-3 

-2 

-  1 

0 

+  1 

+  2 

+  3 

+  4 

i 

1 

2 
4 

8 
16 


4.   5^. 


7.   3-». 


8. 
9. 
10. 


ay 


163.  Properties  of  a'^.  The  graphs  obtained  in  the  pre- 
ceding exercise  indicate  several  properties  of  the  exponen- 
tial function  by  mere  inspection.  In  the  first  place,  all  of 
these  graphs  are  continuous,  unbroken  curves.  To  this  fact 
corresponds  the  following  theorem. 

I.  The  exponential  function  is  continuous  for  all  fiiiite 
values  of  x. 

AVe  shall  not  attempt  to  give  a  formal  proof  of  this  theorem. 

In  the  second  place,  we  observe  that  all  of  the  exponen- 
tial curves  are  situated  entirely  ahove  the  .r-axis.  This  is  a 
consequence  of  the  following  theorem,  which  follows  at  once 
from  the  definition  of  a^. 

II.  The  function  a^  loith  a  positive  base  a,  is  positive  for  all 
values  of  x. 

3Iore  specifically ;  if  a  >  1,  then  a""  >  1  for  x  >  0,  and 
rt^  <  1  for  X  <0;  if  a  <  1,  then  a-"  <  1  for  x  >  0,  a7id  a'^  >  1 
or  X  <  0. 

Let  us  draw  a  line  parallel  to  the  ^--axis  and  above  it. 
The  graphs,  which  we  have  constructed,  indicate  that  such 
a  line  will  intersect  the  curve  in  one  and  only  one  point. 


276  THE   GENERAL   POWER   FUNCTION         [Art.  163 

This  remark  suggests  the  following  theorem  which  we  shall 
not  attempt  to  prove  except  in  this  intuitive  fashion. 

III.  If  y  is  any  positive  number^  there  exists  one  and  only 
one  real  number  x^  such  that  a^  =  y. 

We  observe  further,  if  we  draw  all  of  the  graphs  on  the 
same  sheet  and  referred  to  the  same  axes,  that  they  all  pass 
through  the  point  a;=  0,  ^  =  1.  This  is  due  to  the  follow- 
ing fact  (see  Art.  157)  : 

IV.  For  any  base  «,  we  have  a^  =  1. 

It  is  clear,  botli  from  the  graphs  and  otherwise,  that  a' 
grows  beyond  bound  when  x  grows  beyond  bound  in  the 
positive  direction  ;  but  that  a^  approaches  zero  as  a  limit 
when  X  grows  beyond  bound  in  the  negative  direction,  pro- 
vided that  the  base  a  is  greater  than  unity.  These  facts 
may  be  summarized  as  follows  : 

V.  If  a  y  1,  a^  increases  as  x  increases^  and    lim    a^=  +od. 

VI.  If  a  >  1,  a""  decreases  as  x  decreases^  and    lim    a^=  0. 

The  following  properties  are  not  quite  so  evident  from  the 
graph,  but  follow  immediately  from  the  definition  of  a^  and 
the  index  laws, 

VII.  a^  •  ay  =  a""*"^,  the  addition  formula  for  the  exponential 
function. 

VIII.  —  =  «^~'',  the  subtraction  formula  for  the  exponential 

a" 

function. 

IX.  (a^)"  =  a^^,  the  multiplication  formula  for  the  exponen- 
tial fuyiction. 

The  division  formula  (a-^)i''^  =  a-^''-" 

may  be  regarded  as  being  contained  in  IX,  since  y  in  IX  may 
be  a  fraction,  and  need  not  be  listed  separately.  Similarly 
VIII  may  be  regarded  as  a  consequence  of  VII,  since  y  in 
VII  may  be  positive  or  negative.  However,  VIII  has  so 
many  important  applications  as  to  justify  an  explicit  state- 
ment. 


Art.  164]  DEFINITION   OF   LOGARITHM  277 

X.  If  the  numbers^  .r^,  x,^^  x^,  and  so  on,  form  an  arithmetic 
progression,  the  corresponding  exponentials  a^\  a""",  a'3,  and  so 
on,  form  a  geometric  progression. 

This  follows  from  VII  and  the  definitions  of  arithmetic 
and  geometric  progressions  (Arts.  56  and  59). 

As  a  consequence  of  X,  the  theory  of  geometric  progressions  may  be 
connected  with  the  following  question :  what  values  does  an  exponential 
function  ar'  assume  when  x  assumes  in  succession  the  values  0,  1,  2,  3, 
and  so  on?  In  just  this  way  the  theory  of  arithmetic  progressions  was 
connected  with  the  question  :  what  are  the  values  of  the  linear  function 
clx  +  a  for  X  =  0,  1,  2,  8,  and  so  on?     (See  Art.  56.) 

EXERCISE     LXXV 

Simplify  the  following  expressions  : 

1.   10^  .  1..-  ..       3.  (-_^^j  .        5.  (4^  .  s^y.       7.  [-^-^)  . 

2_  io--io"-\     4.  (■■^±3:^Y.    6.  (4x' .  8x')i.'   8.  :^E2/^. 

10-'  \     9-3^     /  ^  ^  25^ 

164.  Definition  of  logarithm.  If  rt^  is  a  positive  base  differ- 
ent from  unity  wq  know,  according  to  Theorem  III,  Art.  163, 
that  there  exists  a  real  exponent  x  such  that 

(1)  a-=y, 

where  y  is  any  given  positive  number.  This  exponent  x  is 
called  the  logarithm  of  y  with  respect  to  the  base  a,  a  relation 
which  is  expressed  in  symbols  as  follows  : 

(2)  .r  =  log„y. 

Thus,  the  logarithm  of  a  positive  ^lumber  y,  ivith  respect  to  a 
given  base  a,  is  the  exponent  of  the  poiver  to  -which  the  base  a 
must  be  raised  in  order  to  obtain  the  number  y. 

It  should  be  noted  that  we  have  defined  the  logarithms  of  posifive 
numbers  only.  The  question  whether  negative  numbers  have  any  loga- 
rithms need  not  be  discussed  here.  In  the  theory  of  functions,  however. 
logaritlimsof  negative  numbers  are  actually  defined  ;  but  these  logarithms 
of  negative  numbers  are  imaginary. 


278  THE   GENERAL  POWER   FUNCTION        [Art.  165 

EXERCISE     LXXVI 

1.  Express  the  contents  of  the  equation  5^  =  125  in  the  language  of 
logarithms. 

Solution.  This  equation  states  that  the  base  5  must  be  raised  to  the  3d 
power  in  order  to  produce  125.  According  to  the  definition  of  a  log- 
arithm, we  have  therefore 

log,  125  =  3. 

2.  What  are  the  logarithms  of  2,  4,  8,  16,  32,  64,  128  with  respect  to 
the  base  2  ?     Write  out  each  of  these  results  in  symbols  ;  thus  log2  4  =  2. 

3.  What  are  the  logarithms,  3,  9,  27,  81,  243  with  respect  to  the  base 
3? 

4.  What  are  the  logarithms  of  10,  100,  1000,  10,000,  with  respect  to 
the  base  10  ? 

5.  What  are  the  logarithms  of  3,  9,  27,  81,  243  with  respect  to  the  base 
27? 

6.  What  are  the  logarithms  of  1,  \,  \,  ^V'  sT)  2*3  with  respect  to  the 
base  3  ? 

7.  What  are  the  values  of  2^,  3*,  4*,  10^  when  x  is  equal  to  zero? 
What,  then,  is  the  logarithm  of  1  with  respect  to  each  of  the  bases  2,  3, 
4,  10? 

8.  What  is  the  logarithm  of  1  with  respect  to  any  base  a  ? 

9.  What  is  the  logarithm  of  any  number  with  respect  to  itself  as  base? 

10.  Find,  approximately,  to  two  decimal  places,  the  number  whose 
logarithm,  with  respect  to  the  base  2,  is  equal  to  1.5. 

165.   Graph  of  a  logarithmic  function.     The  functions 

(1)  y  =  a^  and  x  =  \og^y 

represent  the  same  relation  between  x  and  y,  merely  written 
in  a  different  form,  just  as  is  the  case  with  the  relations 

y  =z  aP'  and  X  =  ±  V,y- 

In  other  words,  the  two  functions  «="  and  log„?/,  which  are 
inverses  of  each  other  (see  Art.  149),  have  the  same  graph. 
If,  however,  we  prefer  to  write 

(2)  y  =  loga  X 


Art.  lOf)]  PROPERTIES   OF   LOOARTTHMS  279 

SO  that  the  independent  variable  is  denoted  by  x,  as  we  are 
in  the  habit  of  doing,  the  graph  of  (2)  will  be  the  same  as 
that  of  the  equivalent  relation 

(3)  X  =  a^. 

But  this  latter  graph  may  be  obtained  from  the  graph  of 

//=  a"" 

by  the  process  of  reflection  described  in  Art.  149. 

EXERCISE     LXXVII 
Draw  the  graphs  of  the  following  functions: 

1.  ?/  =  logo  X.  5.    1/  =  logio  X. 

2.  y  =  logg  X.  e.  >j  =  log^-  X. 

3-  y  =  log4  ^-  7.  y  =  log^  X. 

4-  y  =  logs  X-  8.  y  =  logi  X. 

3 

166.  Properties  of  logarithms.  The  properties  of  loga- 
rithms follow  at  once  from  those  of  the  exponential  function, 
and  from  inspection  of  the  graphs.  The  most  important  ones 
are  as  follows: 

I.  If  a  is  a  jjositive  number  different  from  unity,  the  function 
log^x  is  defined  for  all  2)ositive  values  of  x,  butnotfor  x  =  0  nor 
for  negative  values  of  x.  It  is  a  continuous  function  for  all 
positive  values  ofx. 

II.  If  the  base  a  is  greater  than  unity,  we  have  log^x  <  0  for 
0  <  a;  <  1,  log^x  >  0  for  x  >  1. 

III.  The  logarithm  of  unity  with  respect  to  any  base  is  zero^ 
that  is 

log^  1  =  0,  since  a^  =1. 

IV.  The  logarithm  of  any  number  with  respect  to  itself  as 
base  is  unity,  that  is, 

log  a  a  =  1,  since  a^  =  a. 

V.  Jf  a  >  1,  then  lim  log^x  =  +  oo  . 


280  THE   GENERAL   POWER   FUNCTION         [Art.  166 

This  is  meielj'  a  re-statement  of  V,  Art.  16o.  For  if  y  =  logaX,  we 
have  a«  =  x  and,  according  to  V,  Art.  163,  x  will  become  infinite  as  y 
grows  beyond  bound.  This  property  is  sometimes  expressed  by  the 
symbolic  equation 

logo  CO   =  CO  . 

VI.  7^  a  >  1,  then  Urn    logaX=  —  co  .      Thus  log^x  is  not 

continuous  in  the  neighborhood  of  .r  =  0, 

This  is  merely  a  re-statement  of  VI,  Art.  163.  For,  if  //  =  logo  a;,  we 
have  ay  =  X  and,  according  to  YI,  Art.  163,  x  will  approach  zero  as  its 
limit  when  y  grows  beyond  bound  through  a  sequence  of  negative  num- 
bers.    This  property  is  sometimes  expressed  symbolically  as  follows : 

logo  0  =  -    CO  . 

VII.  The  logarithm  of  a  product  is  equal  to  the  sum  of  the 
logarithms  of  the  factors. 

Proof.  Let  M  and  N  be  two  jjositive  numbers,  and  let  x  and  y  be 
their  logarithms,  so  that 

X  =  logo  M,   y  =  loga  N, 
or 

M  =  a^,    N  =  nv. 
We  then  have 

MN  =  (fay  =  a^+y,  (Theorem  VII,  Art.  163) 

or,  making  use  of  the  definition  of  logarithms, 

logo  {MN)  =  z  +  ?/  =  logo  M  +  logo  iV, 

and  this  equation  proves  the  theorem. 

VIII.  The  logarithm  of  a  quotient  is  equal  to  the  logarithm 
of  the  dividend  minus  the  logarithyn  of  the  divisor. 

Proof.     Using  the  same  notations  as  in  the  proof  of  VII,  we  find 


and  therefore 


^  =  c^-y, 

N 


logo^  =x-  y  =  logo  M  -  log  N, 


which  was  to  be  proved. 

IX.    The  logarithm  of  the  jo"'  power  of  a  number  M  is  ob- 
tained by  mnltiplging  the  logarithm  of  M  by  p. 


Art.  107]  COMMON   LOGARITHMS  281 

Proof.  If  x  =  logo  M,  we  liave  M  =  a",  and  Mp  =  (a^)p  =  a^*  accord- 
ing to  IX,  Art.  IGo.     Therefore 

loga  .1/''  =  px  =  p  logo  J/. 

X.  The  logarithm  of  the  n""  root  of  a  number  Mis  obtained 
hi/  dividing  the  logarithm  of  M  by  n. 

Proof.     Tliis  theorem  follows  from  IX  by  putting/)  —  -• 

n 

XI.  If  the  numbers  :?-j,  .z^,  .jg,  etc^  are  in  geometric  progres- 
sion^ their  logarithms  will  be  in  aritlimetric  progression. 

Pi:ooK.  This  follows  at  once  from  VII,  if  we  make  use  of  the  defini- 
tion of  an  arithmetic  and  a  geometric  progression. 

Compare  this  theorem  with  theorem  X  of  Art.  163  and  with  the 
fundamental  property  given  in  Art.  160  of  the  power  function. 

EXERCISE  LXXVIII 

1.  If  logio^  =  0.3010,  log,o3  =  0.4771,  and  log,o  5  =  0.6990;  find  log,o  12, 
logio(i),    logio(V-),    log,o</6,    log,ol5.    Iog,o2.5,    logjo  i    logiof. 

2.  Express  in  terms  of  loga/>  and  log^r/,  the  following  quantities : 

l0g„(;/Y),     l0ga(^^'^),     l0g„^/^J,     log^aV^T- 

3.  Prove  the  equation 
X  +  y/x^  -  1 


loga     ^  "  =  2  log„  (X  +  Vx-i  -  1). 

4.  Prove  the  following  statement :  In  order  to  be  able  to  calculate  the 
logarithm  of  any  integer,  it  suffices  to  know  the  logarithms  of  all  prime 
numbers. 

5.  What  functions  are  those  which  have  the  following  property?  If 
the  argument  x  takes  on  a  sequence  of  values  which  are  in  arithmetic 
[>rogres8ion,  the  corre.sponding  values  of  the  function  will  also  be  in 
arithmetic  progression. 

167.  Common  logarithms.  With  scarcely  an  exception, 
the  civilized  nations  ol'  all  times  have  made  use  of  the  deci- 
mal system  for  expressing  numbers,  both  in  the  spoken  and 
in  the  written  language.*  For  this  reason,  the  number  10  is 
especially  well  adapted  to  serve  as  base  for  a  system  of  loga- 

*  It  is  usually  admitted  that  the  prednminance  of  the  decimal  system  over  all 
others  is  due  to  the  fact  that  the  normal  hmnaii  btiiii;  has  ten  finders.  Tliis 
opinion  has  certainly  been  generally  held  since  the  time  of  Aristotle. 


282  THE   GENERAL   POWER   FUNCTION       [Art.  168,  169 

rithms.  Logarithms  with  respect  to  the  base  10  are  usually 
known  as  common  logarithms ;  they  are  also  sometimes  called 
Briggsian  logarithms,  in  honor  of  Henry  Briggs*  (1556- 
1630),  who  constructed  the  first  table  of  common  logarithms. 
For  purposes  of  numerical  calculation,  common  logarithms 
are  by  far  the  most  convenient.  We  reproduce  in  tlie  ap- 
pendix a  four-place  table  of  common  logarithms,  whose  use 
we  shall  now  explain. 

Articles  168-177  may  be  omitted  by  students  who  have  studied  trigo- 
nometry, or  postponed  until  they  take  up  ti'igonometry. 

168.  Characteristic  and  mantissa.    The  positive  integral  powers 

of  10,  sucii  as  10,  100,  1000,  etc.,  the  negative  integral  jaowers  of  10,  such 
as  0.1,  0.01,  0,  0.001,  etc.,  and  the  zero  power  of  10,  which  is  equal  to  1, 
are  the  only  numbers  whose  common  logarithms  are  integers.  The  loga- 
rithms of  all  other  numbers  have  an  integral  and  a  fractional  part. 

The  fractional  part  of  the  logarithm  is  called  the  mantissa,  ivhile  the 
integral  pa7-t  of  the  logarithm  is  known  as  its  characteristic. 

169.  Properties  of  the  mantissa.  We  consider  the  mantissa  and 
the  characteristic  separately  because,  in  practice,  the  method  of  finding 
the  characteristic  of  a  logarithm  is  entirely  different  from  that  employed 
for  finding  its  mantissa.  The  reason  for  this  will  appear  from  the 
following  discussion. 

Let  us  grant  that  we  have  found  out  in  some  way 

(1)  log  1.7783  =  0.2500. 

From  the  theorem  about  the  logarithm  of  a  product,  we  conclude 

log  17.783  =  log  (1.7783  x  10)=  log  1.7783  +  log  10  =  0.2500  +  1 

=  1.2500, 
log  177.83  =  log  (1.7783  x  100)  =  log  1.7783  +  log  100  =  0.2500  +  2 

=  2.2500, 

We  observe  that  the  numbers  1.7783,  17.783,  177.83,  etc.,  contain  the 
same  succession  of  digits,  and  differ  from  each  other  only  in  the  position 
of  the  decimal  point.  Their  logarithms,  on  the  other  hand,  whose 
values  we  have  just  calculated,  differ  from  each  other  only  in  the  value 
of  the  characteristic. 

*  Briggs  was  the  first  Savilian  Professor  of  geometry  at  Oxford.  According 
to  Ball  (see  Ball's  Primer  of  the  Historij  of  Mathematics).  Briggs  was  also  the 
first  to  make  systematic  use  of  the  decimal  notation  iu  working  with  fractions. 


Art.  170]     DETERMINATION  OF  THE  CHARACTERISTIC   283 

Again,  if  we  make  use  of  the  theorem  about  the  logarithm  of  a  quo- 
tient, we  find  from  (1) 

log  0.17783  =  logi—^  =  0.2500  -  1, 
log  0.017783  =  log  ^-^^  =  0-2500  -  2, 


Now,  the  negative  quantities,  whicli  appear  in  the  right  members  of 
these  equations,  are  not  written  in  the  form  wliich  we  ordinarily  use  for 
negative  quantities.  Thus,  for  instance,  we  have  found  the  value  of 
log  0.017783  to  be  0.2500  —  2,  a  result  which  we  should  ordinai-ily  write 
in  the  form  —  1.7500  to  which  it  is  obviously  equal.  If  we  agree  to 
write  every  negative  logarithm  in  this  unusual  form,  as  a  difference 
between  a  positive  proper  fraction  and  an  integer,  thus  making  its  frac- 
tional part  positive,  we  gain  the  advantage  that  the  mantissa  will  be 
the  same  for  any  two  numbers  which  contain  the  same  succession  of 
digits,  even  if  none  of  these  digits  appear  to  the  left  of  the  decimal 
point.  We  avoid,  in  this  way,  the  necessity  of  using  two  different 
tables  of  mantissas,  one  for  numbers  greater  than  unity  and  one  for  num- 
bers less  than  unity. 

Let  us  recapitulate  the  result  of  our  discussion  in  two  formal 
statements. 

I.  We  agree  to  express  the  logarithm  of  any  positive  number  N  in  such  a 
form  that  its  mantissa  shall  be  positive. 

This  can  be  done  w'hether  log  N  is  positive  or  negative,  that  is, 
whether  N  be  greater  or  less  than  unity.  In  the  latter  case,  the  nega- 
tiveness  of  log  iV  is  brought  about  entirely  by  means  of  the  negative 
characteristic. 

As  a  consequence  of  this  agreement,  the  following  statement  will  be 
true  in  all  cases. 

II.  If  two  numbers  contain  the  same  succession  of  digits,  that  is,  if  they 
differ  only  in  the  position  of  the  decimal  point,  their  logarithms  will  hare  the 
same  mantissa  and  icill  differ  only  in  the  value  of  the  characteristic. 

It  is  for  this  reason  that  the  tables  give  only  the  mantissas  of  the 
logarithms  and  that,  in  looking  up  the  mantissas,  we  pay  no  attention  to 
the  position  of  the  decimal  point  in  the  given  number. 

170.  Determination  of  the  characteristic.     The  characteristic 

of  a  logarithm  is  easily  determined  by  inspection.     Its  value    depends 
merely  on  the  position  of  the  decimal  point.     Since  we  have 

10"=  1,  101  _  10,  102  ^  100,  103  =  1000,  etc., 
or  log  1=0,  log  10  =  1,  log  100  =  2,  log  1000  =  3,  etc., 

we  draw  the  followino-  conclusions  : 


284  THE   GENERAL   POWER   FUNCTION         [Akt.  170 

If  1  <  .V<  10,  then  0  <  log  iV<  1.     .-.  log  N  has  the  characteristic  0. 

If  10<iV<100,  then  1  <  log  N<2.  .-.  log  N  has  the  character- 
istic 1. 

If  100<i\^<  1000,  then  2  <logiV<3.  .-.  log  N  has  the  character- 
istic 2. 


If  10*<  A^<  10*+i,  then  A<  log  N  <  k  +  1.  .-.  log  N  has  the  char- 
acteristic k. 

We  may  fornmlate  these  results  as  follows: 

I.  If  k  is  a  positire  integer,  and  if  the  numher  N  lies  between  10*  and 
10*"^S  the  characteristic  of  log  N  is  equal  to  k. 

Since  such  a  number  N  has  k  -\-  1  digits  to  the  left  of  the  decimal 
point,  we  obtain  the  following  rule  : 

II.  If  N  is  any  number  greater  than  1,  the  characteristic  of  its  logarithm 
is  one  less  than  the  number  of  digits  in  its  integral  part. 

The  student  is  advised  to  make  but  little  use  of  this  rule  on  account 
of  its  mechanical  character.  Statement  I  provides  a  better  method 
(less  mechanical  and  easier  to  remember)  for  determining  the  charac- 
teristic. 

It  remains  to  show  how  to  find  the  characteristic  of  log  iVwheu  iV<  1. 

If  .l<iV<l,  then  -l<logiV<0.  .-.  logN  has  the  character- 
istic —  1,  since  the  mantissa  is  positive. 

If  .01<A''<.1,  then  -  2<log7V^<-  1.  .-.  log  N  has  the  charac- 
teristic —  2. 

If  .001  <N  <  .01,  then  -  3  <  log  N  <  -  2.  .-.  log  TV  has  the  charac- 
teristic —  3. 


If  Y7is<  ^  ^ To"*'  *'^^"  ~  ^^  +  ^)<  log  ^^<-  k.  .-.  log  N  has  the 
characteristic  —  (^-  +  1). 

Examination  of  this  table  leads  to  the  following  two  statements, 
either  of  which  may  be  used  to  determine  the  characteristic  of  log  N 
when  iV<  1. 


Ifk  is  a  positire  integer,  and  if  the  number  N  lies  between  ■ —  and 


1 


the  characteristic  of  log  N  is  —  (k  +  1). 

If  N  is  less  than  1,  and  is  expressed  as  a  decimal  fraction  having  k  zeros 
between  the  decimal  point  and  the  frst  significant  figure,  then  the  characteris- 
tic of  the  logarithm  of  N  is  —{k-\-  1 ) . 

In  one  of  our  illustrations  we  had  found 

log  0.01783  =  0.2."j0O  -  2. 


AuT.  171]  USE   OF   TABLE   OF    LOGARITHMS  285 

We  must  never  write  this  in  the  form 

log  0.01783  =  -  2.2500, 

since  only  the  characteristic  is  negative  and  not  the  fractional  part. 
Some  computers  use  the  notation 

log  0.01783  =  2.2500  ; 

but  for  most  purposes  it  is  preferable  to  write 

log  0.01783  =  8.2500  -  10, 
and  similarly 

log  0.1783  =  9.2500  -  10. 

In  other  words,  in  actual  practice,  ice  write  a  positive  characteristic 
10  —  k  in  place  of  the  negative  characteristic  —  k,  and  then  subtract  \0  from 
the  whole  logarithm. 

171.  Arrangement  and  use  of  the  table  of  logarithms.    We 

have  already  mentioned  the  fact  that  the  table  of  logarithms  gives  only 
the  mantissas.  The  characteristics  must  be  supplied  by  the  computer 
by  the  methods  of  Art.  170. 

The  Table  in  the  Appendix  gives  the  mantissa  for  every  number  from 
1  to  999,  to  four  decimal  places.  In  order  to  explain  the  arrangement 
and  use  of  this  table,  we  shall  now  solve  a  number  of  typical  examples. 

Problem  1.     Find  the  logarithm  of  22.1. 

Solution.  To  find  the  mantissa  we  ignore  the  decimal  point.  We 
read  down  the  left-hand  column  of  the  table  (headed  N)  until  we  fii.d 
the  fiist  two  digits  of  our  number,  viz. :  22.  The  numbers  printed  in  the 
same  horizontal  row  with  22  are,  in  order,  the  mantissas  of  the  logarithms 
of  220,  221,  222,  •••,  229,  as  indicated  by  the  number  at  the  head  of 
each  of  the  next  ten  columns.  In  our  case  we  find  the  mantissa,  from 
the  column  headed  1,  to  be  .3444.  Since  22.1  is  between  10  =  10'  and 
100  =  10-,  tlie  characteristic  is  1.     Therefore 

log  22.1  =  1.3444. 

If  the  number  N  contains  more  than  three  digits  its  logarithm  cannot 
be  read  directly  from  the  table.  But  it  may  be  found  by  interpolation. 
We  illustrate  this  process  by  an  example. 

Problem  2.     Find  the  log  222.7. 

Solution.  From  the  table  we  find,  sujiplying  the  characteristics  our- 
selves, 

log  222  =  2.3464 
log  223  =  2.3483 


286  THE   GENERAL   POWER   FUNCTION         [Art.  171 

Tabular  difference  =  0.0019  =  19  units  of  the  fourth  decimal  place. 
Since  222.7  is  ^^  of  the  way  from  222  toward  223  we  add  ^^  of  the  tabu- 
lar difference  to  log  222.     Therefore 

log  222.7  =  2.3464  +  ^V  of  0.0019, 
or  log  222.7  =  2.8464  +  0.0013  =  2.3477. 

It  remains  to  show  how  to  find  the  number  when  its  logarithm  is 
given . 

Problem  3.  Given  log  iV  =  9.3489  -  10.  Find  the  value  of  N  to 
four  significant  figures. 

Solution.  The  characteristic  of  log  iV  is  9  —  10  or  —  1.  Therefore, 
the  number  iV  must  be  between  10-^  =  0.1  and  10°  =  1.  Consequently, 
the  decimal  point  will  precede  the  first  significant  figure  of  N. 

The  mantissa  3489  does  not  occur  in  the  table,  but  it  falls  between 
the  two  tabular  mantissas  3483  and  3502. 
Thus  we  have : 

9.3483  -  10  =  log  0.2230  (from  the  table), 

9.3489  -  10  =  log  iV, 

9.3502  -  10  =  log  0.2240  (from  the  table), 

so  that  N  lies  between  0.223  and  0.224. 

We   observe  that    log  N  lies  j%  of   the  w^ay  from   log  0.223   toward 
log  0.224.     Therefore,  .V  lies  ^%  of  the  way  from  0.223  toward  0.224. 
That  is, 

N  =  0.223  +  j%  of  10  units  of  the  fourth  decimal  place. 
But 

x'y  of  10  units  =  xf  units  =  3^-  units  =  3  units, 

neglecting  j\  which  is  less  than  i. 
Therefore 

iV  =  0.2230  +  0.0003  =  0.2233. 

It  often  happens  that  we  wish  to  subtract  a  logarithm  from  another 

smaller  one.     In  all  such  cases  we  change  the  form  of  the  minuend  by 

adding  and  subtracting  10,  or  some  convenient  multiple  of  10,  as  in  the 

following  example. 

30  34 
Problem  4.     Compute  -^ — 
^        472.3 

Solution.     We  find  from  the  table, 

log  32.34  =  1.5097 
log  472.3  =  2.6742. 


Akt.  172,  173]  EXTRACTION   OF   ROOTS  287 

In  order  to  subtract  the  latter  logarithm  from  the  former,  we  write 
log  32.34  =  11.5097  -  10,* 
log  472.3  =    2.6742 

W  ±i:±I  =    8.8355  -  10 


Hence,  from  the  table,  ^^  =    0.06847 


472.3 
32.34 
472.3 


172.  Extraction  of  roots  by  means  of  logarithms.     Since 

log^x  =  log  x^'P  -  -  log  .V, 

it  is  easy  to  extract  roots  of  any  order  bj'  means  of  logarithms.  If  the 
characteristic  of  log  x  is  not  negative,  no  further  remark  is  necessary. 
If  log  X  is  negative,  we  proceed  as  in  the  following  example  : 

Problem  5.     Compute  by  logarithms  :   V'.o376,  ^.5376,  and  \/.5376. 
Solution,     losr  0.5376  =  9.7305  -  10. 


log  \/.5376  =  ^  log  0.5376  =  » (19.7305  -  20)  =  9.8653  -  10. 


log  V.5376  =  1  log  0..5376  =  ^^9.7305  -  30)=  9.9102  -  10. 


5/""^ 


log  v.5376  =  ^  log  0.5376  =  ^49.7305  -  50)=  9.9461  -  10. 
Therefore,  from  the  table, 


VMlQ  =  .7333,  V.5376  =.8132,  a/.5376  =  .88-32. 

173.  Logarithmic  calculations  which  involve  negative  num- 
bers. We  have  defined  only  the  logarithms  of  positive  numbers.  But 
this  suffices  for  our  purposes.  Clearly,  when  we  compute  a  product  or 
quotient,  its  numerical  value  may  be  found  first,  without  paying  any 
attention  to  the  signs  of  the  various  factors.  Afterwards,  the  proper 
sign  (+  or  —  )  may  be  prefixed  to  the  result  according  as  the  number  of 
negative  factors  is  even  or  odd. 

The  easiest  way  to  keep  a  count  of  the  negative  factors  is  to  use  the 
method  introduced  by  Gauss,  of  writing  the  letter  »  immediately  after 
a  logarithm  which  corresponds  to  a  negative  number.  In  forming  a 
sum  or  difference  of  logarithms,  we  write  an  n  after  the  result  only  if  the 
number  of  separate  logarithms  affected  by  an  n  is  odd. 

Example.     If  iV  =  -  222.7,  we  write 

log  N  =  2.3560  n. 

*  A  computer  with  some  experience  will  refrain  from  actually  writing  the 
logarithm  in  the  form  ll.W)!)?  —  10.  It  i.s  easy  for  him  to  carry  out  the  calculation 
as  thouyh  it  were  so  written. 


288  THE    GENERAL   POWER   FUNCTION     [Akt.  174,  175 

174.  Principles  used  in  logarithmic  calculations.  The  appli- 
cation of  logarithms  to  numerical  calculations  dejjends  upon  the  prin- 
ciples given  in  Art.  166,  esj)ecially  Theorems  VII,  VIII,  IX,  and  X.  As 
a  consequence  of  these  theorems,  multiplication  and  division  of  two 
numbers  may  be  replaced  by  the  simpler  operations  of  addition  and  sub- 
traction of  their  logarithms  ;  and  the  operations  of  involution  and  evolu- 
tion are  very  much  simplified  as  indicated  by  Problem  5  in  Art.  172.  In 
fact,  they  are  reduced  to  simple  multiplications  and  divisions. 

175.  Arrangement  of  the  calculation.  Every  number  and  loga- 
rithm which  occurs  in  a  calculation  sliould  be  properly  labeled,  and  a 
definite  place  should  be  provided  for  it  before  the  calculation  is  actually 
carried  out.  This  is  done  by  making  a  plan  or  a  skeleton  form.  In  mak- 
ing such  a  form,  care  should  be  taken  that  numbers  which  are  to  be  com- 
bined by  addition  or  subtraction  shall  appear  close  together  and  in  the 
same  column.     The  following  example  will  illustrate  this  point. 

Example.  Compute  the  value  of  ;  =  :^  to  four  significant  figures,  if 
X  =  6.320,  //  =  8.674,  z  =  2.851. 

Solution.     AVe  first  make  a  form  as  follows : 

[  X  =  log  X       = 

Given  |  y  =  log  y       = 


[z  =  log  (xy)  = 

To  comj)ute  t  =xy/z  log  c        = 

Result  t  =  log  t        = 

In  this  form  we  have  provided  a  place  (jiroperly  labeled)  for  every  num- 
ber which  will  be  required  in  the  calculation.  We  have  written  log  x 
and  log  y  together  in  the  second  column  because  we  shall  have  to  add 
them  in  order  to  find  log  (.?•//)  for  which  a  place  has  been  provided  just 
below.  We  write  logz  below  this,  because  we  shall  have  to  subtract 
logc  from  log  (xy)  in  order  to  find  \ogt.  Finally  we  provide  a  place 
for  t  which  is  the  number  we  wish  to  compute.  The  necessary  interpo- 
lations are  performed  mentally. 

We  now  fill  out  our  skeleton  with  the  numbers  obtained  from  the 
table. 

[  X  =  6.320  log  X        =  0.8007 

Given     y  =  8.674  log  y        =  0.9382 

[2  =  2.851  log  (x?/)  =  1.7389 

To  compute  /  =  xy/z  log  z        =  0.4550 

/=  19.23  logt        =1.2839 

A  four-place  table  of  mantissas  gives  us  logarithms  correct,  at  most, 
to  the  nearest  unit  of  the  fourth  decimal  place.      Calculation   with   a 


Art.  176]        THE   LOGARITHMIC   OR   GUNTER   SCALE        289 

four-place  Utile  can  there) m-c  give  us  correclhi,  at  mast,  the  first  four  signifi- 
cant Jiyures  of  the  result  sought.  If  greater  accuracy  is  desired,  more 
extensive  tables  must  be  used.* 

EXERCISE  LXXIX 

1.  INIaking  use  of  the  tables,  I'md  log  ;}7l\  log  67.4,  log  729,  log  0.389. 
log  0.00852. 

2.  flaking  use  of  tlie  tables,  find  log  :526o,  log  76.43,  log  8794, 
log  0.04.572,  log  0.005042. 

3.  By  means  of  the  tables,  find  the  numbers  whose  logaritlnns  are 
3.84.52,  1.6807,  8.8906-10,  7.1254-10,  2.2706. 

4.  By  means  of  the  tables,  find  the  numbers  whose  logarithms  are 
3.8906,  9.2411-10,  1.5219. 

5.  Given  a  =  3.157,  b  =  7.291,  c  =  45.73.  Compute  by  logarithms 
the  values  of  ab,  be,  cd. 

6.  With  the  same  values  of  a,  b,  c  compute  ab/c. 

3  j—^r 

7.  With  the  same  values  of  a,  b,  c  compute  A/— . 

8.  Compute  the  volume  of  a  hemispherical  dome  if  its  diameter  is 
150.3  feet.     (Volume  of  a  sphere  of  radius  r  is  Itt/'^.) 

176.  The  logarithmic  or  Gunter  scale.  It  is  possible  to  carry  out 
logarithmic  calculations  graphically.  Since  addition  of  logarithms  corre- 
sponds to  multi2)lication  of  numbers,  we  may  find  the  logarithm  of  a 
product  graphically  by  adding  line-segments,  whose  lengths  are  equal  to 
the  logarithms  of  the  factors. 

But  in  order  to  do  this,  we  must  liave  some  means  for  actually  finding 
a  line-segment  whose  length  shall  be  equal  to  the  logarithm  of  a  given 
number. 

Let  us  take  a  line-segment  of  convenient  length,  say  10  centimeters, 
as  unit  of  length.  In  terms  of  this  unit,  the  whole  distance  (10  centi- 
meters =  100  millimeters)  represents  log  10,  since  the  logarithm  of  10  is 
equal  to  unity.  If  we  count  all  distances  from  the  left-hand  end  of  the 
line,  we  may  label  the  right-hand  end  10  to  indicate  that  this  distance 
represents  log  10.  The  left-hand  end  will  then  be  labeled  1,  because 
log  1  =  0. 

From  the  table  of  logarithms  we  have,  to  two  decimal  places, 

log  1  =  0.00,    log  2  =  0.30,    log  3  =  0.48.    log  4  =  0.60.    log    5  =  0.70, 
^^  log  6  =  0.78,     log  7  =  0.85,    log  8  =  0.90,     log  9  =  0.9.5,     log  10  =  1.00. 

*  See  the  five-place  tables  entitled  Lof/aritlnnic  and  Trir/ouooietric  Tables  by 
WiLczYNSKi  and  Sr.AuoHT,  or  the  six-place  tables  by  ]'>riomikeh,  the  seven-place 
tables  of  Vega,  the  eight-place  tables  of  Bauschingek. 


290 


THE   GENERAL   POWER   FUNCTION         [Art.  177 


Fig.  G3. 


We  mark  the  points  on  our  line-segment  whose  distances  from  the  left- 
hand  end,  measured  in  terms  of  the  whole   line  as  unit,  are  in  order 
equal  to  log  2,  log  3,  log  4,  •••  log  9,  and  label  them  2,  o,  4,  •••  9,  respec- 
tively.    If    the    whole  line-seg- 

1_ I ?       f      ?     ^    ]  ?  ?  y        ment    is    10    centimeters    long, 

these  points  will,  on  account  of 
(1),  be  at  distances  30,  48,  60, 
70,  78,  85,  90,  95  millimeters,  respectively,  from  the  left-hand  end  of 
the  line-segment  (see  Fig.  63). 

A  scale  constructed  in  this  way  is  called  a  logarithmic  scale,  and  its 
usefulness  for  purposes  of  calulation  was  first  pointed  out  by  Edmund 
GuNTER*  in  1620.  It  enables  us  to  find  a  line-segment  equal  in  length 
to  the  logarithm  of  any  number  between  1  and  10.  It  is  easy  to  see 
how,  by  means  of  such  a  scale  and  a  pair  of  dividers,  multiplication  and 
division  may  be  reduced  to  the  simple  graphical  processes  of  adding  and 
subtracting  line-segments. 

177.   The    slide    rule.      Some     years     before      1630,      William 
OuGHTRED  f  noticed  that  the  use  of  .the  dividers  might  be  avoided  by 
constructing  two  equal  logarithmic  scales  (Scales  A  and  B  of  Fig.  64) 
capable  of  sliding  by  each  other,  as  indicated  in  the  figure.^ 

The  use  of  this  simple  bit  of  apparatus  for  the  purpose  of  multiplica- 
tion and  division  will  be  apparent  from  the  following  examples : 

To  multiply  2  by  3.  Place  scale  B  in  such  a  way  that  its  left-hand 
index  (i.e.  the  division  marked  1)  falls  directly  under  the  division 
marked  2  on  scale  A.  Directly  above  the  division  marked  3  on  scale  B, 
we  shall  find,  on  scale  A ,  the  product  which  (of  course)  is  6.  To  justify 
this  pi-ocess  it  suffices  to  note  that  it  is  equivalent  to  adding  the  loga- 
rithm of  3  to  that  of  2. 


A 

1 

1 

r 

•i              4 
1                1 

5 

1 

6 

1 

7      8 
1 

9     10 

1       1 

1 

1 

1 

1 
3 

4 

5 

6 

7 

8 

1       1 
9     10 

B 

Fig.  (54. 


Figure  64  shows  scales  A  and  B  in  the  proper  position  for  the  pur- 
poses of  this  example. 

To  divide  6  by  3.     Under  the  division  6  of  scale  A,  place  division  3  of 


*  Professor  of  a.stronomy  in  Gresham  College,  London  (1581-1626). 
t  OuGHTRED  (1,")7.5-1()()0)  was  a  fellow  of  King's  College,  Cambridge. 
t  Oughtrcd's  instruments  were  described  iu  publications  of  William  Foster, 
one  of  his  pupils,  in  1632  and  1633. 


Art.  177] 


THE   SLIDE   RULE 


291 


scale  B.    .Over  tlie  division  1  of  scale  ]j  we  shall  find  the  quotient  (|=  2) 
on  scale  .1  (cf.  Fig.  (!l). 

The  instrument  actually  in  use,  the  Mannheim  slide  rule,  is  a  slight 
amplification  of  the  one  just  described  (cf.  Fig.  65).  It  has  four  scales, 
usually  denoted  by  A,B,  C,  D,  respectively,  the  scales  A  and  D  being  on 
the  rule,  and  B  and  C  on  the  slide. 


Fk;.  (W. 


The  scale  A  is  composed  of  two  logarithmic  scales  such  as  that  of 
Fig.  63,  so  that  its  right-hand  end  might  be  labeled  100,  since  log  100  =  2. 
On  most  slide  rules,  however,  the  first  principal  division  on  scale  A  after 
9  is  not  labeled  10,  as  in  Fig.  63,  but  1,  the  next  one  is  not  labeled  20, 
but  2,  and  so  on  to  the  last  one,  which  is  again  labeled  1  instead  of  100 
or  10.  Thus,  the  two  halves  of  scale  A  are  exact  copies  of  each  other. 
This  is  done  for  precisely  the  same  reason  that  the  mantissas  only  are 
printed  in  our  tables  of  logarithms.  The  slide  rule  also  makes  use  of  the 
mantissas  only.  The  characteristics,  or  what  amounts  to  the  same  thing, 
the  position  of  the  decimal  point  in  the  result,  must  be  obtained  by 
inspection  or  by  special  rules. 

Scale  B  is  on  the  upper  edge  of  the  slide,  in  direct  contact  with  scale 
A  on  the  rule,  and  is  an  exact  copy  of  scale  A.  These  two  scales 
together  may  be  used  for  multiplication  and  division  as  explained 
above. 

Scale  D  is  on  the  lower  part  of  the  rule.  It  is  a  single  logarithmic 
scale,  from  1  to  10,  of  the  same  length  as  the  combined  two  scales  of  A. 
The  logarithm  of  any  number  is  therefore  represented,  on  scale  D,  by 
a  distance  twice  as  great  as  that  which  represents  the  logarithm  of  the 
same  number  on  scale  A.  It  follows  from  this  that  the  number  which 
is  found  on  scale  A,  vertically  above  any  number  of  scale  D,  is  the 
square  of  the  latter.  Any  number  on  scale  D,  on  the  other  hand,  is  the 
square  root  of  the  number  vertically  above  it  on  scale  A . 

Scale  C  is  on  the  lower  edge  of  the  slide,  in  direct  contact  with  slide 
D  on  the  rule.  It  is  an  exact  copy  of  scale  D.  These  two  scales  to- 
gether may  be  used  for  multiplication  and  division,  according  to  the 
same  rules  which  hold  for  scales  A  and  B. 

Besides  these  four  scales,  the  slide  rule  is  sxipplied  with  a  runner 
(cf.  Fig.  65),  which  is  useful  in  performing  compound  operations,  and 
also  in  comparing  two  scales  (such  as  A  and  D),  which  are  not  in  direct 


292  THE   GENERAL   POWER   FUNCTION         [Art.  178 

contact  with  each  other.  The  runner  was  made  a  permanent  feature  of 
the  slide  rule  by  Mannheim  in  1851.* 

It  often  happens,  in  manipulating  the  slide  rule,  that  the  result  is  to 
be  sought  opposite  a  number  of  the  slide  which  falls  outside  of  the  scale 
on  the  rule.  In  such  cases,  we  may  shift  the  slide,  bringing  the  right- 
hand  index  to  the  place  which  the  left-hand  index  occupied  previously, 
and  read  off  the  result  as  before.  For  such  a  shift  has  no  influence  on 
the  mantissa,  since  it  merely  amounts  to  dividing  the  result  by  10.  On 
the  Mannheim  rule,  this  shifting  of  the  slide  may  be  avoided  by  work- 
ing with  scales  A  and  B  rather  tlian  with  C  and  D.  Scales  C  and  D, 
however,  have  the  advantage  of  greater  accuracy. 

If  the  slide  be  withdrawn  entirely,  it  will  be  found  to  have  three  other 
scales  on  its  reverse  side,  two  of  which  are  labelled  S  and  T.  These  are 
scales  of  logarithmic  sines  and  tangents,  respectively,  and  may  be  used 
for  calculating  such  products  as 

c  sin  A ,  c  tan  A . 

The  middle  scale  on  tlie  reverse  side  is  used  for  finding  the  value  of 
the  logarithm  of  a  number,  and  is  important  if  we  wish  to  compute  a 
power  of  a  number  with  a  complicated  fractional  exponent. 

For  more  complete  information  concerning  the  slide  rule,  we  must 
refer  to  the  manuals  which  are  usually  presented  to  the  purchaser  of 
such  an  instrument.!  Cheap  slide  rules,  especially  constructed  for  the 
beginnei-,  may  now  be  obtained  of  all  dealers  under  the  name  Student's 
or  College  Slide  Rule.  Engineers  and  coniputers  use  the  slide  rule  so 
extensively  that  the  student  will  find  it  advisable  to  make  himself 
familiar  with  the  instrument  by  actual  use. 

The  Mannheim  slide  rule,  which  we  have  described,  admits  of  three- 
figure  accuracy.  In  some  (exceptional)  cases,  results  correct  to  four 
decimal  places  may  be  obtained  by  its  use.  The  Thacher  and  Fuller 
slide  rules,  more  complicated  instruments,  but  constructed  on  essentially 
the  same  principles,  admit  of  far  greater  accui-acy. 

178.  The  general  notion  of  a  scale.  The  logarithmic  scale, 
which  is  used  in  the  slide  rule,  and  the  familiar  scale  of  inches  on  a  yard- 
stick are  two  special  instances  of  the  general  notion  of  a  scale.  In  both 
of  these  cases  the  scales  are  straight.  One  of  them,  the  scale  of  inches, 
is  also  uniform,  that  is,  the  divisions  of  the  scale  are  numbered  in  such 
a  way  that  the  points  labeled  1,  2,  3,  4,  etc.,  are  at  equal  distances  from 
each  other.     The  logarithmic  scale  is  straight  but  not  uniform.     The 

*  Amkdke  Mannheim  (ISiU-lPOf)),  a  distinguished  geometer  of  recent  times. 
The  nnuier  \v,u\  Iiowever  been  used  occiisionally,  long  before  Manuheim,  by  a 
iiumbei'  of  English  mathematicians. 

t  See  also  Raymond's  Plane  Surveying. 


Art.  179]      LOGARITHMS  OF  DIFFP^RENT  SYSTEMS  293 

scale  of  degrees  on  a  graduated  circle  is  iiiiifonn  but  not  straight.  The 
scale  of  hours  on  a  sun-dial  is  neither  uniform  nor  straight. 

These  illustrations  will  suffice  to  explain,  even  without  a  formal  def- 
inition, what  is  meant  by  a  scale  in  general.  The  essential  cliaracterisdc 
of  a  scale  is  that  it  establishes  a  one-to-one  correspondence  between  the 
points  of  a  straight  or  curved  line  on  the  one  hand  and  the  numbers  of 
a  certain  set  on  the  other  hand.  Moreover,  this  correspondence  is  such 
that  if  a  point  describes  the  line  continuously  in  a  certain  direction  w  ith- 
out  ever  going  backward,  the  corresponding  numbers  will  grow  continu- 
ously from  the  lowest  number  of  the  set  to  the  highest. 

Such  scales,  both  straight  and  curved,  both  uniform  and  non-uniform, 
are  exceedingly  useful  for  purposes  of  measurement  and  graphical  com- 
putation and  are  used  extensively  in  practice. 

179.  Relation  between  the  logarithms  of  two  different  sys- 
tems. Let  a;  be  a  positive  number.  If  we  choose  a  positive 
number  a  as  base,  let 

(1)  \o^^x  =  p. 

If  we  clioose  a  second  positive  number  h  as  base,  let 

(2)  log6  x=q. 

We  wish  to  investigate  the  relation  which  exists  between 
the  logarithms  of  the  same  number  x  taken  with  respect  to 
the  two  different  bases,  a  and  h. 

From  (1)  and  (2)  we  have,  by  the  definition  of  logarithms, 

(3)  a;  =  aP,  x  =  b'' 
and  therefore 

which,  on  account  of  the  division  formula  for  the  exponen- 
tial function  (see  IX,  Art.  163),  may  be  written  in  either 
of  the  two  forms 

a''  =  h  or  a  =  b^. 
By  the  definition  of  logarithms  we  have  therefore 

logg  b  =  '   and  logj,  a  =  i, 
q  '  p 


294  THE   GENERAL   POWER   FUNCTION         [Art.  179 

whence,  by  multiplication, 

(4)  logj,  a  .  log„  b  =  l. 
This  result  may  be  expressed  as  follows. 

Theorem  I.  If  a  and  h  are  two  positive  numbers,  the  loga- 
rithm of  a  un'th  respect  to  the  hase  h  is  the  reciprocal  of  the 
logarithm  of  h  with  respect  to  the  hase  a. 

Let  us  now  take  the  logarithm,  with  respect  to  the  base 
J,  of  both  members  of  the  first  equation  of  (3).  We  find 
(see  Art.  166,  Theorem  IX) 

\og^x=p\ogi,a. 

But,  according  to  (1),  p  is  equal  to  log„a^.     Consequently, 
we  find 

(5)  logft  X  =  log„  X  •  logs  a. 

On  account  of  (4)  this  may  be  written,  more  conveniently, 
as  follows : 

Theorem  II.  Equation  (6)  enables  us  to  compute  the 
logarithm  of  x  with  respect  to  the  neiv  base  b,  when  the  loga- 
rithm of  X  and  the  logarithm  of  b  with  respect  to  the  old  base 
a  are  known. 

Suppose  we  actually  have  at  our  disposal  a  table  of  com- 
mon logarithms.  Then  we  know  log^Q  x  for  all  values  of  rr, 
and  we  may  also  find  from  the  table  the  value  of  logjg  5, 
where  b  is  any  positive  number. 


Theorem  III.      Consequently  the  formula 


enables  us  to  construct  a  table  of  logarithms  with  respect  to  any 
base  b.     For  this  purpose  it  is  only  necessary  to  divide  all  of 

*  This  means  los^a  x  divided  by  loga  b,  and  not  loga  \,\<  which  would  be  equal 
to  logo  a;  — logo  &. 


Art.  180]  STANDARD   LOGARITHMIC   CURVE  295 

the  logarithms  of  the  common  system  hy  one  and  the  same  divi- 
sor, namely  by  the  common  logarithm  of  the  new  base. 

More  generally,  formula  (6)  tells  us  the  following : 

Theorem  IV.  If  the  logarithms  of  all  numbers  are  known 
with  respect  to  some  particular  base  a,  the  logarithms  of  all 
nu7nbers  ivith  respect  to  any  other  base  b  may  be  found  by  di- 
viding all  of  the  logarithms  of  the  first  system  by  one  and  the 
same  divisor,  namely  by  the  logarithm  of  the  second  base  tvith 
respect  to  the  first. 

It  is  clear,  then,  that  the  knowledge  of  one  system  of 
logarithms  is  sufficient  to  give  us  complete  information 
about  all  other  systems  of  logarithms.  This  becomes  even 
more  evident  if  we  express  our  last  theorem  geometrically, 
as  follows : 

Theorem  V.  If  the  graph  of  the  function  log^x  has  been 
constructed,  the  graph  of  any  other  logarithmic  function,  log^  x, 
may  be  obtained  from  it,  by  either  diminishing  or  increasing 
all  of  the  ordinates  of  the  first  graph  in  the  same  ratio. 

This  is  so,  because  division  by  log^  b  will  decrease  all  ordi- 
nates in  the  same  ratio  if  log,,  Z>  >  1.  It  will  increase  all 
ordinates  of  the  graph  in  a  fixed  ratio  if  log„  ^'  <  1 . 

EXERCISE  LXXX 

1.  Making  use  of  the  table  in  the  Appendix,  compute  the  logarithms 
of  2,  3,  4,  5,  6,  7,  8,  9,  10  with  respect  to  the  base  5.  How  would  you 
proceed  if  you  wished  to  construct  a  four-place  table  of  logarithms  with 
respect  to  the  base  5  ? 

2.  Draw  the  graph  of  y  =  log.,x.  What  nuist  you  do  to  this  graph 
iu  order  to  obtain  the  graphs  oi  y  =  log^  x,    ij  =  logg  x,   y  =  log^-  x? 

3.  Construct  a  scale  for  tlie  function  x\  for  values  of  x  between  0 
and  1,  making  use  of  the  computed  values  of  x-  for  x  —  0.0,  0.1,  0.2,  ••• 
0.9,  1.0. 

180.   Selection  of  a  standard  logarithmic  curve.     Whatever 

may  be  the  base  a,  the  curve  obtained  as  a  graph  of 

y  =  loga  X 


296  THE   GENERAL   POWER   FUNCTION         [Art.  181 

will  always  pass  through  the  point  x  =  1,  ?/  =  0,  since 
log„l  =  0  for  all  bases.  (See  Fig.  Q6.^  But  although  these 
curves  all  have  this  point  in  common, 
each  of  them  will  have  a  different  tangent 
at  this  point.  We  shall  select  as  a  stand- 
ard logarithmic  curve  that  one  whose 
tangent  at  the  common  point  makes  an 
angle  of  45°  with  tlie  a;-axis,  on  account 

Fig.  60.  ^  ' 

of  tlie  central  cliaracter  of. this  curve  as 
compared  with  all  of  the  others.  The  question  which  we 
sliall  have  to  settle  is  this :  what  is  the  base  of  that  par- 
ticular system  of  logarithms  whose  graph  is  the  standard 
logarithmic  curve  ?  We  shall  denote  tliis  base  by  e  and 
speak  of  the  corresponding  logarithms  as  natural  logarithms. 

181.   The  derivative  of  the  logarithmic  function.     In  order 

to  be  able  to  answer  the  question  raised  in  Art.  180,  we 
must  first  show  how  to  compute  the  slope  of  the  tangent  at 
any  point  of  a  logarithmic  curve.  The  same  argument 
which  was  used  in  Art.  87,  and  which  was  there  applied 
only  to  integral  rational  functions,  gives  us  the  following 
result : 

Let  f  (x^  he  a  continuous  function,  and  let  us  construct  its 
graph  hy  putti7ig 

y=f(^)- 

If  this  curve  has  a  tangent  at  that  one  of  its  points  whose 
coordinates  are  x  and  y,  the  slope  of  the  tangent  will  he  ohtaitted 
hy  evaluating  the  limit : 

(1)  /'  (,0  =  lim  /(^  +  /0-/(x)  , 

A->o  h 

This  limit  is  called  the  derivative  off(x'). 
In  our  case  we  have 

/(a;)=  log^z, 
and  therefore 

f(x-\-h)=]oQ;,(x+h). 


Art.  181]  DERIVATIVE   OF    LOGARITHM  297 

Consequently 

^2)    f(^  +  h~)-fix)  ^  \og„(ix  +  h)-  log, a; ^  1  ,^^^,^  fx  +  h\ 
h  h  h     '"""V    X    J 

where  the  last  two  members  are  equal  ou  account  of  VIII, 
Art.  166. 

We  wish  to  evaluate  the  limit  which  (2)  ap[)roaches  when 
X  has  any  definite  fixed  value  while  h  approaches  zero  in  any 
manner  whatever.  Since  the  function  log^x  is  not  con- 
tinuous for  x  =  0  (see  Art.  166,  Theorem  VI),  we  shall,  on 
this  account,  assume  that  the  fixed  v^lue  assigned  to  x  is  not 
zero. 

We  first  rewrite  (2)  as  follows : 

(3)  /(2L±^l^££)  =  l,og,(i+^^). 

Since,  in  evaluating  this  limit,  x  is  to  be  I'egarded  as  a  fixed 
number  different  from  zero,  we  may  introduce 

(4)  ^  =  t 

X 

as  a  new  variable  in  place  of  h.  As  7i  approaches  zero,  t  will 
also  approach  zero.     But  from  (4)  we  find 

h  =  xt 
and  (3)  now  becomes 

h  X  t     ' 

or,  on  account  of  IX,  Art.  166, 

/(.+A)-/(.)^i      ^^,)j. 

k  X 

If  now  we  make  use  of  (1),  we  find  the  expression 

(5)  /(a:)  =  ^limlog„(l  +  0' 

X  /->o 

for  the  derivative  of  the  function  log^  x. 


(6)  J\x')  =  -\og^e 

X 


298  THE   GENERAL   POWER   FUNCTION        [Art.  181 

1 
Let  us  assume  provisionally  that  (1  +  ^ '  actually  ap- 
proaches a  definite  finite  limit,  different  from  zero,  when  t 
approaches  zero  in  any  manner  whatever,  and  let  us  denote 
this  limit  by  the  letter  e.  Since  the  function  log^a;  is  con- 
tinuous in  the  neighborhood  of  any  finite  positive  value  x  =p 
(see  Art.  166,  Theorem  I),  we  may  write 

lim  logo  a;  =  logojt).         (See  Art.  96.) 
We  shall^  therefore,  Jifid  from  (5)  the  following  expression 

(6) 

for  the  derivative  of 

or,  what  amounts  to  the  same  thing,  for  the  slope  of  the  tangent 

of  the  curve  y  —  log„  x  at  that  one  of  its  points  which  has  the 
abscissa  x. 

Let  us  apply  this  theorem  to  that  point  of  the  curve  for 
which  x=l  and  therefore  y  =  0.*  We  find  the  following 
result :  The  slope  of  the  line  which  is  tangent  to  the  graph  of 
logo  a;  *^  ^^^  point  a;  =  1,  y  =  0,  is  equal  to  log^  e  where 

1 
(7)  e  =  lim  (1  +  0^ 

In  order  that  this  tangent  may  make  an  angle  of  45°  with 
the  a;-axis,  its  slope  must  be  equal  to  unity.  Therefore  in 
this  case  log„e  must  be  equal  to  1,  and  this  is  so  if  and  only 
if  a  =  e. 

Consequently,  the  standard  logarithmic  curve,  ivhose  tangent 

at  the  point  a;  =  1,  ?/  =  0  makes  an  angle  of  45°  with  the  x-axis, 

corresponds    to    that    system  of  logarithms  whose   base    is    the 

number 

1 

e  =  lim  (1  +  ty. 

The  number  e  is  called  the  natural  or  Napierian  base,  and 
the  corresponding  system  of  logarithms  is  called  the  system 

*  This  is  the  point  common  to  all  logarithmic  curves.     (See  Art.  180.) 


Art.  182]  THE   NUMERICAL  VALUE  OF  e  299 

of  natural  logarithms.     The  notation  In  x  is  sometimes  used 
for  the  natural  logarithm  of  x. 

182.   The  numerical  value  of  e.     It  is  a  rather   difficult 

matter,  and  quite  beyond  the  scope  of  this  book,  to  prove  that 

1 
(1  +  ty  actually  approaches  a  definite  finite  limit  when  t  ap- 
proaches zero.     To  prove  this  it  is  necessary  to  show  that 

1 
(1  +  t)'-  will  approach  the  same  limit  when  t  approaches  zero 
throuoli  a  sequence  of  positive  values  as  when  t  approaches 
zero  through  negative  values,  that  the  limit  will  be  the  same 
whether  t  approaches  zero  continuously  or  by  a  series  of 
jumps.  If,  however,  we  grant  the  existence  of  the  limit  we 
can  easily  gain  a  very  fair  idea  as  to  what  the  value  of  it 
will  be,  by  allowing  t  to  approach  zero  in  some  particularly 
convenient  fashion. 

Let  us  then  think  of  t  as  assuming  in  succession  the  values 

.111  1 

'    2'    8'    4'    "■    n     "" 
We  shall  then  have 

(1)  e  =  lim  (1  +  0'  =  lim  f  1  +  -  Y  • 
We  actually  find 

(1  +  1)1  =  2,    (1  +  . 1)2  =2.25,  etc. 

By  using  a  five-place  table  of  logarithms  we  easily  find  (to 
four  significant  figures) 

(1  +  ^)10  =  2.594,    (l  +  3io)ioo  =  2.704. 

The  true  value  of  e  to  eight  decimal  places  is 

(2)  g  =  2.71828123. 

We  may  also  obtain  this  value  as  follows.  Let  us  expand 
f  1  +  -  j  by  means  of  the  binomial  theorem.  (See  Art.  88.) 
We  find 


300  THE   GENERAL   POWER   FUNCTION         [Art.  182 

/-J    ,  1Y  =  1   I  y^l  ■  n{n-l)  1      n{n-l)(n  —  2}  1 
\        7iJ  In  1  •  2      w^  1  •  2  •  3  n^ 


n(n-l)(n-2)(n-d)  1 

1.2.3.4  71^ 


1-1  (i-^Yi-?) 


=  1  +  1+1.-'   ■         l.:i.8 


l_lYl-2\,i_ 


+ rnnri — -+ 

12    3 

As  w  grows  beyond  bound,  -,  -,  -,  etc.  all  approach  zero, 

n    n    n 

and  it  seems  })lausible  that  we  should  find 

limfl+-Y  =  l  +  l+  — + ~ + +  ••• 

n-^K        nj  1.21.2.31.2.3.4 

where  the  law,  according  to  which  the  terms  on  the  right- 
hand  side  are  formed,  is  evident.  Thus  we  find  the  follow- 
ing formula  for  e, 

(3)         e=l  +  l  +  -l^  + 1 + 1 — ^+  .... 

^^  11  .2      1.2.3      1  .2-3.4 

We    have    not    actually  proved  this    formula.     For  as   w 

12    8 
grows  beyond  bound  some  of  the  fractions  -,  — ,  -,  etc.  will 

71    71    n 

have  numerators  which   also   grow  beyond   bound,  and  the 

number     of     factors     which     occur     in     sucli     i)roducts     as 

(1 )(1  — -)(1  —  -]  will  also  grow  beyond  bound.     Con- 

V        nj\         nj\         n)  _  ^  /       IV 

sequently  it  is  not  at  all  certain  tliat  ( 1  H —  j  actually  ap- 
proaches the  right  member  of  (3)  as  a  limit.  It  is  possible 
to  prove,  however,  that  this  is  actually  the  case. 


Art.  is;}]  EXPONENTIAL    EQUATIONS  301 

Formula  (3)  enables  us  to  compute  the  value  of  e  with  great  rapidity 
to  as  many  decimal  places  as  may  be  desired.  Thus,  we  have  (to  five 
decimal  places) 


1  +  1^  =  2.00000 

^^=0.50000 

^        —  0  1  fififiT 

1.2.3~          ^^ 

^            _  0  oilfiY 

1  .2.3.4-  "-"^^^^ 

0  0099^ 

__.       .     ._                —  W.uUOOO 

1.2-3.4.5 

1                      _  A  AA1  on 

r 

.2.3.4.5.6-"-"^^'^^ 

1               _  0  ooo^n 

1-2 

.3.4.5.0.7 

^                 —  0  nono'^ 

1.2.3 

.4.5. 6.7. s-"-"^^^"^ 

2.71829 

a  result  which  agrees  with  (2)  to  within  one  unit  of  the  fifth  decimal 
place. 

183.  Exponential  equations.  An  equation,  some  or  all  of 
whose  terms  are  exponential  functions  of  the  unknown  quan- 
tity, is  called  an  exponential  equation.  We  have  no  general 
method  for  solving  exponential  equations,  but  the  method  of 
trial  and  error  explained  in  Art.  99  will  usually  enable  us  to 
find  approximate  values  for  the  unknown  quantity. 

Tliere  are  two  cases,  however,  in  which  the  problem  may  be 
reduced  to  that  of  algebraic  equations. 

Case  I.     The  equation  is  of  the  form 

where  a,  5,  c,  .••,  /,  m,  n,  •••  are  constants,  and  where  the  ex- 
ponents/(x),  g{x),  7i(2-),  ..-,  </)(./•),  V^O'),  %(.*•),  ".are  all 
either  constants  or  rational  functions  of  x. 

If  we  equate  the  logarithms  of  both  members  of  (1),  we  find 

(2)  /  (:c)  logio  a  +  g(^x)  log^^  ^^  -f-  •  •  • 

=  <^(.»)  logio  I  +  >/r(.r)  logiy  m  -h  x(x)  It'Sio  «  +  ••• 


302  THE   GENERAL    POWER   FUNCTION         [Art.  183 

and  this  is  a  rational  equation,  which  may  be  solved  by  the 
methods  of  Chapter  VI,  Art.  145.  This  method  consists  in 
reducing  (2)  to  an  equivalent  integral  rational  equation, 
which  may  then  be  solved  by  the  methods  of  Chapters  II  to  V. 

Thus,  the  equation 

23x521-1  _  45131+1 

becomes,  if  we  take  the  logarithms  of  both  members, 

3  x  log 2  +  (2  X  -  1)  log5  -Tjx  log  4  +  (x  +  1)  log  3 

where  common  logarithms  are  meant.     This  may  be  written 

(3  log  2  +  2  log  5  —  5  log  4  —  log  3)x  —  log  3  +  log  5, 
whence 

^  ^ log  3  +  log  5 ^ 

3  log  2  +  2  log  5  —  5  log  4  —  log  3 ' 


or 

X 


log3  +  log5  ^  0.4771  +  0.6990         ^  _  ^  gg^g 

2  log5  -  7  log 2  -  log 3      1.3980  -  2.1070  -  0.4771 


Case  II.     The  equation  is  of  the  form 

(3)  Aa"-f''-^  +  ^a'"-i'-^^-^>  +  (7a("-2'/('>  +  •••  +  La-^^-''^  +  M=  0. 

where  f(^x^  is  a  rational  function  of  x. 
In  this  case  we  put 

(4)  «/(-)  =  ^, 

thus  reducing  (3)  to  an  algebraic  equation  for  ^,  namely 

(5)  A2/"  +  Btr-'  +  -'  +  Li/  +  M=0. 

Lety^  be  one  of  the  n  roots  of  (5).     Then  we  find  from  (4), 
taking  logarithms  of  both  members, 

/(a:)  log  a  =  log  .Vi- 
and this  is  an  algebraic  equation  for  x. 

Thus,  the  equation,  in  which  c  denotes  a  given  number, 

(6)  1(6=^  +  e-»^)  =  c, 

may  be  reduced  to  the  form  (3)  by  multiplying  both  members  by  2  e". 

It  then  becomes 

e2i  +  1  =  2  e'c 
or 

ga»  _  2  e*c  +  1  =  0. 


Art.  184]  CALCULATION  OF  LOGARITHMS  803 

If  we  put 
(7)  e*  =  /y  or  x=  logey, 

we  find  a  quadratic  equation  for  ij,  namely 
which  gives,  when  solved  for  y, 


y  =  c  ±  Vc^  -  1, 
and  therefore,  on  account  of  (7), 


X   =  log,  ((•  ±  \/c2  -  1). 

EXERCISE  LXXXI 

Solve  the  following  equations.     Give  the  numerical  results  correct  to 
four  significant  figures. 

1.  2'  =  64.  3.   4»='-2*  =  64.  5.   S'^-^^+i  =  \. 

2.  2'-*  =  5.  4.    .5»'-8*  =  2^.  6.   2^  =  14. 
7.    l;r^+5  =  14^+7.                                    10.   a  3'^  +  /;  3^  +  c  =  0. 


8.  log  Va;  —  21  +  ^  log  x  =  1.  11.   a  ;>  -\-  J>  b-^  -{■  c  —  0. 

9.  K2^  -  2-*)=  f-  12.   Aa^  +  Ba'  +  C  =  0. 

184.  The  calculation  of  a  table  of  logarithms.  We  ap- 
proached the  theory  of  logarithms  by  way  of  the  index  laws. 
But  this  was  not  the  path  pursued  by  the  first  inventors  of 
logarithms,  John  Napier  (1550-1617)  and  Jobst  Burgi 
(1552-1632).*  In  fact,  the  notation  x"  was  not  in  use  in 
their  time  and  consequently  the  index  laws  were  not  avail- 
able to  them,  although  in  a  certain  sense  they  were  probably 
well  known  even  then.f 

Both  Napier  and  Biirgi  observed  that  the  numerical  opera- 
tions involved  in  multiplication  are  much  more  burdensome 
than  those  required  in  addition.     They,  therefore,  sought  a 

*  Napier  was  of  Scotch  and  Biirgi  of  Swiss  nationality.  Biirsi's  discovery 
of  logarithms  was  unquestionably  independent  of  Napier's  and  was  made  at 
about  the  same  time.  But  Napier's  book  Miriftci  Lof/aritkmonmi  canonis 
descriptio,  containing  an  account  of  his  method,  was  published  in  1614,  six  years 
earlier  than  Biirgi's  Arithmetisdte  and  Grometrische  Pror/ress-Tahulen.  For 
an  account  of  the  history  of  logarithms  consult  Cajori  in  the  Aiaerican 
Mathematical  Monthly,  Vol.  20  (19i;i). 

t  David  Eugknk  Smith,  Tht:  Law  of  Exponents  in  the  Works  of  the  Sixteenth 
Centunj.  Napier  Tercentenary  Memorial  Volume.  Royal  Society  of  Edinburgh. 
1915. 


304  THE   GENERAL   POWER   FUNCTION         [Art.  184 

method  of  reducing  multiplication  to  addition,  and  both  of 
them  accomplished  this  purpose  independently  by  the  follow- 
ing scheme : 
Let 

\i-)  d-^i   0-c^i   (l^>,   d^i    •••,   (In-,    •" 

be  a  sequence  of  numbers  in  geometric  progression,  and  let 

(2)  5^,  62,  ^3,  ?>4,  ...,  J„,  ... 

be  a  second  sequence  of  numbers  which  are  in  arithmetic 
progression.  Moreover,  let  us  think  of  these  two  sequences 
as  being  in  correspondence,  so  that  to  a^  corresponds  6„. 
Then  to  the  product  of  two  numbers  «i  and  a;^  of  the  first 
sequence  will  correspond  the  sum  of  the  corresponding  two 
numbers  hi  and  b/^  of  the  second  sequence.  Consequently, 
if  we  actually  have  two  such  sequences  worked  out  we  may 
multiply  a^  by  %  as  follows.  Find  the  numbers  h^  and  5^  of 
the  second  sequence  which  correspond  to  a,-  and  %  respec- 
tively, add  bi  and  J^.,  and  then  find  the  number  of  the  first 
sequence  which  corresponds  to  b^  -\-  b/..  This  will  be  the 
product  of  tti  and  a^.-. 

Of  course  this  scheme  is  for  us  merely  an  application  of 
Theorem  XI  of  Art.  166.  In  fact,  the  numbers  Jj,  b^-,  ••• 
will  be  the  logaritlims  of  tfj,  a^-,  "•  with  respect  to  some  base. 
But  for  Napier  and  Biirgi,  tliis  correspondence  between  the 
terms  of  a  geometric  and  an  arithmetic  progression  was  not 
a  theorem,  but  served  as  a  definition  for  logarithms. 

It  is  easy  enough  to  find  sucli  correspondences.  Thus, 
the  numbers 

(3)  1,  10,  100,  1000,  10,000,  ... 
form  a  geometric  progression,  anil  the  numbers 

(4)  0,1,2,3,4,... 

form  an  arithmetic  progression  of  the  kind  described  in 
(1)  and  (2).  But  in  order  that  this  correspondence  may 
actually  be  useful  for  purj)oses  of  calculation,  the  terms  of 
each  of  the  two  progressions  should  be  much  closer  together 


Akt.  184]  CALCULATION   OF   LOGARITHMS  305 

than  they  are  in  the  two  progressions  (3)  and  (4).  We 
therefore  insert  any  convenient  number  of  geometric  means 
between  any  two  terms  of  (3),  and  just  as  many  arithmetic 
means  between  the  corresponding  two  terms  of  (4).  The 
easiest  way  to  do  this  is  to  insert  one  mean  of  each  kind  at  a 
time,  since  tliis  may  be  accomplisluMl  l)y  merely  extracting  a 
square  root. 

Thus,  tlie  geometric  mean  between  tlie  first  two  terms  of 
(3)  is  V10=  3.1623.  (See  Art.  61.)  The  arithmetic  mean 
between  the  first  two  terms  of  (4)  is  1.5.  (See  Art.  57.) 
Our  two  progressions  now  read  as  follows : 


(5) 

1, 

3.1623, 

10, 

31.623, 

100, 

316.23, 

1000, 

(^!) 

0, 

0.5, 

1, 

1.5, 

2 

2.5, 

3, 

In  order  to  have  the  terms  of  the  progression  still  closer 
to  each  other,  we  use  the  geometric  mean  between  the  first 
two  terms  of  (5)  namely  V3.162o  =  1.7783,  and  the  arith- 
metic mean  between  the  first  two  terms  of  (6)  namely  0.25. 
Our  two  progressions  now  become 

(7)  1,     1.7783,     3.1623,     5.6234,     10,  ... 
and 

(8)  0,     0.2500,     0.5000,     0.7500,      1,  .... 

In  our  notation  this  signifies  that 

log  1.7783  =  0.2500,  log  3.1623  =  0.5000,  etc. 

It  is  clear  how,  by  continuing  this  process,  a  table  of 
logarithms  will  result.  To  be  sure,  this  will  not  yet  be  in 
convenient  form,  since  the  numbers  are  arranged  in  geomet- 
ric progression  instead  of  being  spaced  equally.  But  from 
the  logarithms  obtained  in  this  way,  the  logarithms  of  inter- 
mediate numbers  may  afterward  be  obtained  by  interpola- 
tion, thus  finally  enabling  us  to  find  the  logarithms  of  the 
equally  spaced  numbers  1.0,  1.1,  1.2,  1.3,  etc. 

There  are  other,  far  more  convenient,  methods  for  calculat- 
ing a  table  of  logarithms,  and  some  of  these  will  be  explained 


306  THE   GENERAL   POWER   FUNCTION     [Arts.  185-187 

later  on,  although  a  complete  proof  of  these  other  methods 
is  beyond  the  scope  of  this  book. 

185.  Applications  of  logarithms.  In  all  extensive  numeri- 
cal calculations  which  involve  multiplication  or  division,  the 
introduction  of  logarithms  is  advisable.  Trigonometry  is 
full  of  illustrations  of  this  remark.  Another  field  in  which 
logarithmic  calculation  is  almost  indispensable  is  offered  by 
the  problems  of  compound  interest,  some  of  which  we  shall 
now  discuss. 

186.  Simple  interest.  When  a  capitalist  lends  out  money, 
he  usuall}^  charges  the  borrower  a  fee  which  is  called 
interest.  The  sum  loaned  is  called  the  principal ;  the  princi- 
pal plus  the  interest  accrued  at  the  end  of  any  period  is 
called  the  amount  due  at  that  time.  The  rate  is  said  to  be 
i2%  annually  if  interest  is  charged  at  the  rate  of  R  cents  per 
year  for  every  dollar  of  the  principal. 

If  f  P  is  the  principal,  and  R  the  rate,  the  interest  at  the 

PR 

end  of  one  year  will  be  ^- At  the  same  rate,  the  interest 

nPR 

at  the  end  of  n  years  will  be  $ ,  and  the  amount  due  at 

^  100 

the  end  of  n  years  will  be 

(1)  ^=^-+^=^(1  + 

In  what  follows  we  shall  use  the  letter  r  to  stand  for 
^/lOO.     Then  (1)  assumes  a  simpler  form,  namely 

(2)  A^  =  P(\  +  nr). 

187.  Compound  interest.  If  the  borrower  pays  his  interest 
annually,  the  formula  just  derived  gives  a  correct  result  for 
the  total  amount  which  he  should  return  to  the  lender.  But 
if  he  wishes  to  pay  nothing  until  the  n  years  have  passed, 
this  formula  should  be  modified.  For,  by  retaining  the 
various  installments  of  interest  as  they  become  due  from 
year  to  year,  he  is  depriving  the  lender  not  only  of  the  use 


Art.  187]  COMPOUND   INTP:REST  307 

of  the  principal,  but  also  of  the  interest  which  each  installment 

might  have  earned  for  the  lender  in  the  meantime.     Account 

is  taken  of  this  circumstance  in  computing  compound  interest. 

The  interest  at  7v%    annually   on   a   jirincipal  of  8P   is 

PR 

— — -•     Thus  the  amount  at  the  end  of  the  first  year  is 

100  -^ 

where  again  we  put 

^  R 
**     lOO' 

During  the  second  year  we  regard  JLj  as  the  principal.  The 
amount  at  the  end  of  the  second  year  will  be 

A^  =  A^-\-  A^r  =  A^Q.  +  r). 

If  we  substitute  for  A^  the  value  just  found,  this  becomes 

Similarly,  during  the  third  year  we  regard  A^  as  principal, 
and  Und 

A,^F(l  +  ry 

as  the  amount  due  at  the  end  of  the  third  year.  Finally  we 
find  the  formula 

(1)  A^  =  P(l  +  ry. 

for  the  amount  due  at  the  end  of  n  years  on  a  principal  of  $P 
at  the  rate  of  R%  a  year  compound  i7itere8t,  if  the  interest  is 
compounded  annually. 

Formula  ("2)  of  Art.  18G  shows  that  the  amounts  due  at  the  end  of 
one,  two,  three,  etc.,  years  form  an  arithmetic  progression  at  simple 
interest.  Formula  (1)  of  the  present  article  shows  that  they  form  a 
geometric  progression  when  compound  interest  is  charged. 

Formula  (1)  may  also  be  used  to  solve  the  following  prob- 
lem. What  sum  must  be  invested  now,  allowing  compound 
interest  at  the  rate  of  R(fo  annually,  so  that  the  amount  at 
the  end  of  n  years  shall  be  a  given  sum  ? 

Let  P  be  the  unknown  sum  to  be  invested,  and  let  A^  be 


308  THE   GENERAL   POWER   FUNCTION         [Art.  188 

its  amount  at  the  end  of  n  years.  Then  A^  and  P  are  con- 
nected by  equation  (1)  and  A^  is  to  be  regarded  as  a  given 
quantity  ;  consequently  we  tind 

(2)  P=^„(l+rr" 

by  solving  (1)  for  P ;  the  value  of  P  is  called  the  present 
value  of  a  sum  A^  which  is  to  be  paid  at  the  end  of  n  years, 
allowing  compound  interest  ?it  R  %. 

188.  Annuity.  A  fixed  sum  paid  annually  is  called  an 
annuity.  Let  us  compute  the  amount  A  of  an  annuity  of 
•fa  per  year  which  is  allowed  to  accumulate  for  n  years  with 
compound  interest  at  72  % . 

This  problem  is  important  in  such  cases  as  the  following.  A  corpora- 
tion (perhaps  a  life  insurance  company)  makes  a  contract  with  a 
person  X,  promising  to  pay  him  a  certain  sum  of  money  at  the  end  of 
twenty  years,  in  return  for  certain  fixed  sums  called  premiums  which  are 
to  be  paid  by  X  to  the  corporation  annually.  The  corporation  invests 
its  money  at  R  per  cent.  In  order  to  be  able  to  fulfill  its  contract,  the 
company  must  know  how  to  calculate  the  sum  of  all  the  payments  made 
by  A'  plus  the  interest  on  these  payments. 

If  the  first  annual  installment  a  is  due  at  the  end  of  the 
first  year,  it  earns  interest  during  w  —  1  years ;  therefore  its 
amount  will  be  a(\  -\-  r)""^.  The  second  installment  earns 
interest  during  w— 2  years;  its  amount  will  therefore  be 
a(l  -I-  r)"~^.  The  last  installment  will  earn  no  interest. 
Thus  the  total  amount  will  be 

A  =  a(l-j-r)"-i-fa(l-|-r)"-2  4.  . . .  j^  a{\  ■\- r')  ^  a^ 

or  if  we  write  the  terms  in  inverse  order, 

J.  =  a  +  a(l  +  r)  -F  a(l  -f-  r)2  -t-  •  •  •  -f  a(l  -}-  ry-^. 

The  terms  in  the  right  member  form  a  geometric  progres- 
sion of  n  terms,  the  common  ratio  being  1  -|-  r.  Therefore, 
the  sum  is  (Art.  60) 

(1)    A^a^-^Sl^^a\^(l±ir  =  'L^a  +  rr-^. 

1  —  (1  +  r)  —r  r 


akt.  i«!)]  compound  interest  309 

Formula  (1)  enables  us  to  compute  the  amount  A  of  an  annuity 
of  ^  a  per  year  which  is  allowed  to  accumulate  for  n  years  at  R 
per  cent  compound  interest. 

It  is  often  necessary  to  compute  the  present  value  of  an 
annuity  of  '^a  ?i  year  to  continue  for  71  years,  allowing  com- 
pound interest  at  the  rate  of  R  per  cent, 

Tims,  a  person  wishes  to  pay  a  corporation  noiv  a  lump  sum  to  be 
returned  to  him  as  an  annuity  of  f  500  per  year  for  ten  years.  The 
corporation  must  know  how  to  compute  the  lump  sum  which  is  adequate 
for  this  purpose. 

The  present  value  of  ^a  due  one  year  hence  is  a(\  +  r)"^. 
(See  equation  (2),  Art.  187.)  The  present  value  of  $a  due 
two  years  hence  is  •a(l  +  r)"^,  etc.  The  present  value  of 
$a  due  71  years  hence  is  a(l  +  r)~".  Therefore,  the  total 
present  value  of  this  annuity  is 

P=  a{\  +r)-i  +  a(l  +r)-^+  •••  +  a(l  +r)-". 

This  is  a  geometric  progression  of  n  terms,  with  the  common 
ratio  (1  +  r)~i.     Tlierefore,  we  have 

-(1+^- 


P  =  a(l+r)-i[i^ 


(1  +  0-M 


1  —  1/(1  4-r) 


or 

1  +  r      r/(l  +  r)         r 

If  n  grows  beyond  bound,  (1  +  r)~"  tends  towards  zero, 
since  1  +  r  is  greater  than  unity.  Therefore,  the  present 
value  of  a  perpetual  annuity  (called  a  perpetuity^  of  -fa  per 
year  is  a/r(\  —  0)  =  a/r. 

189.  Interest  compounded  more  than  once  annually.  We 
found  in  Art.  187  the  formula 

A  =  P(\  +  r)",  where  r  =  - — -, 
^  lUO 

for  the  amount  due  at  the  end  of  n  years  on  a  principal  P  at 
the  rate  of  R  per  cent  a  year,  if  the  interest  is  compounded 
annually.  If  the  interest  is  compounded  more  than  once  a 
year,  this  formula  must  be  modified. 


310  THP:    general   power   function         [Art.  189 

Let  us  suppose  that  the  interest  is  to  be  compounded  t 
times  a  year,  and  let  us  speak  of  the  f""  part  of  a  year  as  a 
period.     The  interest  at  the  end  of  the  lirst  period  will  be 

P-;  therefore  the  amount  at  the  end  of  the  first  period  is 

(1)  A  =  ^  +  i'^  =  ^(i  +  0. 

During  the  second  period  we  regard  A^  as  the  principal. 
The  interest  on  this  principal    earned    during   the    second 

period  will  be  A-^-  so  that  the  amount  at  the  end  of  the 
second  period  will  be 

A,  =  A,  +  A/-  =  A,(l  +  '^ 

which,  on  account  of  (1),  gives 

Since  n  years  contain  nt  periods,  we  find  finally  the  formula 

(2)  ^  =  p(l  +  0"    where  r=^, 

for  the  amount  due  at  the  end  of  n  years  on  a  principal  P  at 
the  rate  of  R  per  cent  annually.,  if  the  interest  is  compounded 
t  times  a  year. 

In  this  formula  we  may  think  of  t  as  growing  beyond 
bound.  As  a  consequence  the  period  will  approach  zero  as 
a  limit,  and  we  find  the  following  result.     The  formula 

(3)  A  =  P  lim  f  1  +  ^ 

will  give  the  amount  due  at  the  end  of  n  years  on  a  principal 
P  at  the  rate  of  R  per  cent  annually,  if  tlie  interest  is  com- 
pounded instantaneously  or  continuously. 

But  we  are  in  a  position  to  actually  evaluate  this. limit. 
Let  us  put  t  =  kr.     Since  r  is  a  fixed  number,  k  will  then 


Art.  100]  THE   COMPOUND   INTEREST   LAW  311 

grow  beyond  hound  as  t  becomes  infinite.      We  may  there- 
fore write  in  place  of  (3), 

■\\knr 


(4)  A  =  P  lim    1  + 

But  we  have  found  (see  Art.  182), 

lim('l  +  ^Y=., 
x->i'  \        kJ 

so  that  (4)  becomes 

(5)  A  =  Pe-\ 

Till s  i8  the  final  formula  for  the  amount  due  at  the  end  of  n 
years  on  a  principal  P  at  the  rate  of  R  per  cent  annually,  if 
the  interest  is  compounded  continuously. 

We  observe  that  the  amount,  as  given  by  (5),  is  an  ex- 
ponential function  whose  base  is  e,  the  exponent  being  the 
product  of  the  annual  rate  and  the  number  of  years. 

EXERCISE    LXXXII  * 

1.  Find  the  amount  of  f  157.38  for  7  years  at  '^>\  ^c  compound  interest. 

2.  How  much  money  must  I  put  into  the  bank  at  3  %  compound  in- 
terest, so  that  the  amount  may  be  $  500  at  the  end  of  five  years? 

3.  What  will  be  the  amount  of  f  10,000  after  ten  years,  at  4^0  com- 
pound interest,  if  the  interest  is  compounded  annually?  if  the  interest 
is  compounded  semiannually?  quarterly? 

4.  What  is  the  value  of  an  annuity  of  !f  1000  for  a  term  of  twenty 
years,  if  money  is  worth  3  %  per  annum  ? 

5.  How  long  will  it  take  a  sum  of  money  to  double  itself  at  5  9?  com- 
pound interest  per  year  ? 

190.  The  compound  interest  law.  Formula  (5)  of  Art.  189 
has  many  applications  besides  the  one  already  noted.  The 
essential  thing  about  this  formula  is  that  it  represents  the 

*  In  many  of  these  examples  four-place  tables  of  logarithms  are  not  sufficient 
to  give  results  accurate  to  the  nearest  cent.  In  solving  such  examples,  the 
student  should  l)e  satisfied  with  an  approximate  result  if  he  has  no  other  tables 
at  his  disposal  tlian  those  given  in  the  Appendix  of  this  book.  Otlierwise  he 
should  use  some  of  the  more  extensive  tables  mentioned  in  the  footnote  on 
page  289. 


312  THE   GENERAL   POWER   FUNCTION         [Art.  191 

amount  ^  as  a  function  of  w,  the  number  of  years  (P  and  r 
being  regarded  as  constants),  and  that  this  function  has  the 
special  property  that  its  rate  of  growth  at  any  instant  is  pro- 
portional to  its  own  magnitude.  For  that  is  the  way  in 
which  the  amount  due  on  a  loan  grows  when  the  interest  is 
compounded  instantaneously.  Whenever  we  have  a  relation 
of  this  kind  between  two  variables  x  and  ?/,  such  that  the 
rate  of  change  of  y  is  proportional  to  y  itself,  then  y  will  be 
connected  with  x  by  means  of  an  equation  of  the  form 

(1)  y  =  ae'"' 

where  a  and  b  are  constants ;  that  is,  y  will  be  an  exponential 
function  of  x.  On  account  of  this  connection,  the  relation 
between  two  variables  expressed  by  an  equation  of  the  form 
(1)  is  frequently  called  the  compound  interest  Imv^  a  name  in- 
troduced by  Lord  Kelvin  (1824-1907),  a  famous  British 
physicist.  The  importance  of  this  law  lies  in  the  fact  that  it 
occurs  so  frequently  in  the  applications,  some  of  which  we 
shall  now  explain.  In  making  these  applications  it  should 
be  remembered,  however,  that  5,  the  coefficient  of  x  in  (1), 
may  be  either  positive  or  negative.  If  a  and  h  are  both 
positive,  as  in  the  case  of  Art.  189,  y  is  an  increasing  function 
of  X  and  the  constant  increase  in  log  y  which  corresponds  to 
an  increase  of  one  unit  in  x  is  called  the  logarithmic  increment. 
If  a  is  positive  while  h  is  negative,  ?/  is  a  decreasing  function 
of  a;,  and  the  decrease  in  log  y  which  is  caused  by  a  unit  in- 
crease of  x  is  called  the  logarithmic  decrement. 

191.  Dampened  vibrations.  If  a  weight  is  attached  to  one 
end  of  a  string,  whose  other  end  is  attached  to  a  fixed  sup- 
port, an  impetus  given  to  the  weight  will  cause  it  to  oscillate 
about  its  position  of  equilibrium.  This  simple  piece  of  ap- 
paratus is  called  a  pendulum.  Let  the  motion  of  the  pendu- 
lum take  place  in  a  vertical  plane.  The  displacement  of  the 
pendulum  from  its  position  of  equilibrium  at  any  instant 
may  be  measured  by  the  angle  which  the  string  makes  at 
that  moment  with  a  vertical  line.     The  larsfest  value  of  this 


Art.  192]  PRESSURE   IX   THE   ATMOSPHERE  313 

angle  during  one  comjjlete  oscillation  is  called  the  amplitude 
of  the  oscillation.  Owing  to  friction  and  resistance  of  the 
air,  the  amplitude  of  the  vibration  gradually  decreases.  Ob- 
servation shows  that  the  amplitudes  of  successive  vibrations 
of  the  pendulum  form  (very  approximately)  a  decreasing 
geometric  progression.     We  may  therefore  write 


k-l 


(1)  A  =  A,e- 

where  t  represents  the  time,  expressed  in  seconds  or  some 
other  unit  of  time,  which  has  elapsed  from  the  beginning  of 
the  motion,  where  Aq  represents  the  amplitude  (expre.ssed  in 
radians  or  degrees)  of  the  vibration  at  the  beginning  of  the 
motion,  and  where  A  represents  the  amplitude  after  t  time 
units  have  passed. 

The  coefficient  of  t  in  the  exponent  has  been  written  in  the 
form  —  k^  to  emphasize  the  fact  that  it  is  negative.  This 
must  be  so  to  make  each  amplitude  smaller  than  the  preced- 
ing one.  The  actual  value  of  this  coefficient  depends  upon 
the  amount  of  friction  and  resistance.  If  k  is  large,  for  in- 
stance if  the  pendulum  is  swinging  in  water,  the  vibrations 
will  cease  very  soon.  If  k  is  small,  it  will  take  a  long  time 
before  the  pendulum  comes  to  rest. 

Strictly  speaking,  if  formula  (1)  be  regarded  as  absolutely  true,  it 
will  always  take  an  infinite  time  to  produce  absolute  rest.  For,  accord- 
ing to  (1),  A  cannot  become  equal  to  zero  for  any  finite  value  of  t.  But 
when  the  continually  increasing  quantity  k-t  l)ecoines  large  enough,  the 
corresponding  value  of  A  will  become  so  small  as  to  become  immeasur- 
able. When  this  condition  has  been  reachetl,  the  pendulum  may  be  re- 
garded as  being  at  rest,  for  practical  purposes,  since  its  motion  has 
become  imperceptible. 

Formula  (1)  is  applicable  to  other  cases  of  dampened  vi- 
bration. The  case  of  a  pendulum  swinging  in  air  is  merely 
a  particular  case  of  dampened  vibrations. 

192.  Variation  of  density  and  pressure  in  the  atmosphere. 
It  is  a  familiar  fact  that  the  air  is  less  dense  on  the  top  of  a 
mountain  than  at  sea  level.  It  is  easy  to  see  why  this  should 
be  so,  since  the  air  at  higher  levels,  by  its  weight,  helps  to 


314  THE   GENERAL   POWER   FUNCTION         [Art.  193 

compress  the  air  which  is  below  it.  For  the  same  reason  the 
pressure  (as  measured  by  a  barometer)  is  less  at  greater  alti- 
tudes than  at  sea  level. 

The  following  formula  due  to  Halley  (1656-1742),  a 
famous  English  astronomer,  very  nearly  represents  the  facts 
as  obtained  by  observations  at  various  altitudes.  Let  p^  be 
the  density  of  air  at  sea  level  (compare  Art.  44  for  definition 
of  density),  and  let  p  be  the  density  of  air  at  an  altitude  of  h 
meters  above  sea-level.  According  to  Halley's  formula  we 
shall  have 


(1)  p  =  PqB 


8000 


The  pressure  of  the  atmosphere  at  any  height  may  be 
measured  in  pounds  per  square  inch,  or  may  be  expressed  by 
means  of  the  barometer  in  terms  of  millimeters  of  mercury. 
Since  the  pressure  is  proportional  to  the  density,  we  may  also 
write 

(2)  P=Poe~^. 

where  Pq  is  the  pressure  at  sea  level  (15  pounds  per  square 
inch,  corresponding  to  a  height  of  the  barometer  of  760 
millimeters),  and  where  p  is  the  pressure  at  a  height  of  h 
meters  above  sea  level. 

193.  Transmission  of  light  by  imperfectly  transparent  me- 
dia. When  light  passes  through  a  medium  like  air,  water, 
or  glass,  some  of  the  light  is  absorbed,  although  most  of  it  is 
transmitted.  The  amount  of  light  absorbed  depends  upon 
the  nature  of  the  medium  and  its  thickness.  Even  glass 
ceases  to  be  transparent  if  it  is  thick  enough. 

Let  L  be  the  intensity  of  the  light  transmitted  by  a  sheet 
of  glass  X  millimeters  thick.     We  shall  have 

(1)  L  =  L,e-^'^ 

where  i^  is  the  intensity  of  the  incident  light,  that  is,  the 
intensity  of  the  light  before  any  absorption  has  taken  place, 
and  where  k"^  is  a  coefficient  whose  value  depends  upon  the 
quality  of  the  glass. 


Art.  194]  COOLING   BODIES  315 

Let  us  take  the  intensity  of  the  incident  light  as  unit  of  intensity. 
Suppose  that  a  pane  of  ghiss  ;}  millimeters  thick  absorbs  2%  of  the  inci- 
dent light.     Then  we  shall  have  Z^,  =  1,  and  L  —  0.98  for  x  =  3,  so  that 

0.98  =  e-3*'. 
Consequently  we  have 

-  3  ^-2  log  e  =  log  0.98, 
whence 

^,  ^  _  logos  ^  _  9.9912  -  10  ^     0,0088  ^ 
31oge  1.3029  1.3029 

so  that  we  find  for  this  quality  of  glass  the  formula 

for  the  percentage  of  light  transmitted  by  a  pane  x  millimeters  thick. 

194.  Cooling  bodies.  If  a  warm  body  is  placed  in  a 
niedium  whose  temperature  is  kept  fixed  and  which  is  cooler 
than  the  body,  the  latter  will  cool  off  at  a  rate  proportional 
to  the  difference  between  its  temperature  and  that  of  the  sur- 
rounding medium.  This  law  of  cooling,  due  to  Newton, 
may  be  expressed  by  the  formula 

0  =  0^  +  (0^-e,)e-''\ 

where  Oq  is  the  temperature  of  the  medium,  ^j  is  the  original 
temperature  of  the  body,  6  its  temperature  after  t  time  units 
have  passed,  and  F  is  a  coefficient  whose  value  depends  upon 
the  material  of  wliich  the  body  is  composed  and  upon  the 
nature  of  the  surrounding  medium. 

EXERCISE   LXXXIII 

1.  Assuming  Halley's  formula,  at  what  height  will  the  pressure  of 
the  atmosphere  be  just  one  half  of  what  it  is  at  sea  level  ? 

2.  A  body  of  temperature  55"  C.  is  cooling  off,  surrounded  by  air 
whose  temperature  is  15°  C.  After  eleven  minutes  the  temperature  of 
the  body  was  found  to  be  25''  C.  Find  the  value  of  k-  and  write  out 
Newton's  formula  for  this  body. 

3.  According  to  the  formula  found  in  Example  2,  what  time  is  re- 
quired for  the  body  to  cool  off  5°  ? 

-4.  A  dampened  vibration  begins  with  an  amplitude  of  10  centimeters. 
After  nine  minutes  the  amplitude-  is  found  to  be  only  one  centimeter. 
Express  by  a  formula  the  change  which  takes  place  in  the  amplitude 
in  the  course  of  time. 


316 


THE   (GENERAL    POWER    FUNCTION         [Akt.  195 


195.    Semi-logarithmic  paper.      Wlienever  we  have  two  variables 
X- and  y  connected  by  an  exponential  equation, 

(1)  y  =  ae'"^, 

as  in  Arts.  190-19-1,  we  may  construct  the  graph  of  this  equation  as  in 
Art.  162.  The  resulting  graph  is,  of  course,  an  exponential  curve.  We 
may,  however,  convert  the  graph  into  a  straight  line  by  the  following 
device : 

Let  us  take  the  logarithms  of  l)oth  memliers  of  (1).     We  find 

(2)  log  y  =  log  a  +  log(e''^)  =  log  a  +  hx  log  e. 
If  now  we  put 

(3)  A'  =  X.    Y  =  log  y,   h  log  e  =  M,  log  a  =  B, 
equation  (2)  may  be  written 

(4)  Y  =  MX  +  B. 

If  we  regard  X  and  Y  as  the  coordinates  of  a  point  in  the  plane,  the 
locus  of  equation  (4)  is  a  straight  line  of  slope  M  which  intersects  the 
I'-axis  in  the  point  (0,  B).  (See  Art.  54.)  Thus,  the  transformation 
which  consists  in  putting 


(5) 


A'  =  X,    Y  =\ogy 


transforms  the  exponential  curve  which   is  the  locus  of  equation   (1)    into  a 
straight  line. 

In  order  to  actually  carry  out  this  transformation  we  might  proceed 
as  follows:  If  (x,  y)  are  two  numbers  which  satisfy  equation  (1),  we 
put  X  equal  to  x  and  compute  Y  =  log  y.     Then  we  plot  the  point  whose 

coordinates  are  A' and  F  instead 
of  the  point  (.c,  y).  But  an 
easier  way  of  accomplishing  the 
same  result  is  to  provide  the 
3^-axis  with  a  logarithmic  scale 
such  as  is  used  on  the  slide  rule. 
(See  Arts.  170  and  177.)  The 
point  of  the  //-axis  which  corre- 
sponds to  any  number  y  on  this 
scale  will  then  be  at  a  distance 
from  the  origin  which  is  equal 
to  log  y.  If  the  a:-axis  is  pro- 
vided with  an  ordinary  uniform 
scale,  we  can  lay  off  our  x-co6r- 
^  '^  0  1  s  'J  10  dinates  as  usual.  Ruled  paper 
Fig.  67.  on    which    the    rulings   are    ;ir- 


Art.  196] 


LOGARITHMIC   PAPER 


317 


riuiged  in  this  way  is  called  se mi-hxiarithnic  papfr.  and  is  particularly  well 
adapted  for  the  grapliic  representation  of  exponential  eciuations  of  tlie 
form  (1).  On  semi-logarithmic  paper,  tlie  graph  of  such  a  relation  is 
a  straight  line.     Figure  07  represents  a  sheet  of  semi-logarithmic  paper. 


196.  Logarithmic  paper.  If  both  axes  are  provided  with  loga- 
rithmic scales,  instead  of  one  of  the  axes  only,  and  if  the  paper  is  ruled  in 
accordance  with  these  scales,  we  obtain  a  sheet  of  logarithmic  paper.  See 
Fig.  08.  This  is  particularly  adapted  for  the  graphic  representation  of 
relations  such  as 

(1)  7/   =    flX*, 

in  which  a  power  function  occurs.  For,  if  we  take  the  logarithms  of 
both  members  of  (1),  we  find 

(2)  log  y  =  log  a  +  log  a*  =  log  a  +  I:  log  x. 
Consequently,  if  we  put 

(3)  X  =  log  X,    Y  =  log  y,   B  =  log  a,    M  =  k, 
eipuition  (2)  becomes 

(-1)  Y  =  MX  +  B, 

the  graph  of  which  is  a  straight  line.  Thus,  on  lor/nrithmic  paper  wJiere 
both  sets  (if  rulinr/s  are  made  on  a  logarithmic  scale,  the  graph  of  any  power 
function  is  a  straight  line. 

If,  as  a  result  of  a  series  of 
observations,  pairs  of  corre- 
sponding values  of  x  and  ^  are 
given,  and  if  a  preliminary  in- 
spection of  these  values  sug- 
gests that  the  relation  between 
X  and  y  may  be  of  the  foi'm 
of  a  power  function,  y  =  ax^, 
it  will-  be  advisable  to  plot  the 
points  on  logarithmic  paper. 
If  the  i-esulting  graph  is  a 
straight  line,  we  can  easily  find 
the  values  of  M  and  B  and 
afterward,  from  (8),  the  values 
of  a  and  k.  If  the  relation 
seems  to  be  of  the  form  of  an 

exponential  function,  it  will  instead  be  advisable  to  use  semi-logarithmic 
paper. 


■) 

6 

1) 

1 

Fiu.  (jH. 


318  THE   GENERAL   POWER   FUNCTION         [Art.  196 

EXERCISE   LXXXIV 

On  semi-logarithmic  paper  plot  the  following  relations  : 

1.  y  =  10^  4.   y  :=  10-^.  7.    ?/  =  2^  10.   y  =  2"'. 

2.  y  =  102^  5.    y  =  10-2^.  8.    y  =  3\  11.    y  =  3"*. 

3.  y  =  108^.  6.    7/  =  10-3'.  9.   y  =  5^  12.   y  =  5"^ 

13.  Plot  Halley's  formula  (Art.  192)  for  the  density  on  semi-loga- 
rithmic paper,  h  being  expressed  in  kilometers  and  putting  p^  =  1,  that 
is,  using  the  density  at  sea  level  as  unit  of  density. 

14.  Plot  the  relations  obtained  in  Examples  2  and  4  of  Exercise 
LXXXIII  on  semi-logarithmic  paper. 

1  _1  2  _2 

15.  Plot  the  relations  y  =  x-,  y  —  x  ^,  y  =  x^,  y  =  x  ^  on  logarithmic 
paper. 


CHAPTER   IX 

LINEAR  FUNCTIONS  OF  MORE  THAN  ONE  VARIABLE. 
LINEAR  EQUATIONS  AND  DETERMINANTS  OF  THE 
SECOND   AND    THIRD    ORDER 

197.  Functions  of  two  variables.  A  variable  z  is  said  to  be 
a  function  of  two  independent  variables^  x  and  y,  if  to  defi- 
nitely assigned  tmlues  of  x  and  y  there  correspond  definite 
values  of  z.  Such  relations  are  usually  indicated  by  equa- 
tions of  the  form 

(1)  z=f{x,y'), 

where  /  may  be  replaced  by  other  letters  sucli  as  JP,  ^,  i/r, 
and  so  on.  The  equation  (1)  may  be  read  z  is  equal  to  the 
/-function  of  x  and  y. 

The  area,  .1,  of  a  rectangle,  whose  base  is  h  units  long  and  whose  alti- 
tude contains  h  units  of  length,  is  bh  square  units.  Thus  A  =  M  is  a 
function  of  b  and  h. 

EXERCISE    LXXXV 

1.  Tf  /(x,  y)^  x  +  y  +\,  find  the  values  of  /(O,  0),  /(I,  0)./(0,  1), 
/(I,  1),/(1,  ^-2). 

2.  If  F  (x,  //)  =  li  -  3  X  +  7  y  +  xy,  find  the  values  of  F  (0,  0),  F  {\,  1), 
F{%  -1),  F(_:5,2). 

3.  If  </)(r,  y)  —  '2  X  —  y  +  '^,  find  some  values  of  x  and  //  for  which 
<f>(x,  y)  becomes  equal  to  zero. 

4.  If  f(r,  y)  =x^  +  xy  +  y\  prove  that  /(-  x,  -  y)  =f(x,  y),  that 
f(x,  y)  =f(y,  .r),  and  that  f(hx,  ky)  =  k-f(x,  y)  where  k  denotes  any 
number  whatever. 

5.  If  f(x,  fj)  =  x^  +  y^,  what  will  be  the  value  of  /(-  x,  -  y),  of 
/(y.   x),  and  oi /(kx,  ky)? 

6.  Express  the  area  of  a  triangle  as  a  function  of  its  base  and  alti- 
tude. 

319 


320     LINEAR  EQUATIONS  AND  DETERMINANTS      [Art.  108 

7.  Express  the  volume  of  a  rectangular  parallelepiped  as  a  function 
of  the  length  of  three  mutually  perpendicular  edges. 

8.  Express  the  volume  and  the  total  surface  of  a  right  circular  cylin- 
der as  functions  of  the  altitude  and  the  radius  of  the  base. 

198.  Linear  functions  of  two  variables.  Ami  function  of 
the  two  variables  x  and  y/,  which  cayi  he  expressed  m  the  form 
ax  +  by  +  <?,  ivhere  a,  b,  and  c  are  constants,  is  called  a  linear 
function  of  x  and  y. 

Thus,  X  +  y/  +  1,  3  x  +  I  //  —  1,  .r  —  y  —  2,  —  2  x  +  //  +  7  are  linear 
functions  of  x  and  y. 

When  we  compute  the  values  of  a  linear  function  of  x  and 
//,  we  find  that  there  are  many  pairs  of  values  for  x  and  //  to 
which  corresponds  the  same  value  of  the  function. 

Tims,  let/(.r,  ij)  =  x  +  //  +  1.     We  find 

/(O,  4)  =/(!,  :})  =/(2.  2)  =/(:',,  1)  =/(!.  0)  ../(.l,  -  1)  =  5. 

If  we  plot  the  points  (0,  I),  (1,  o),  (2,  2),  etc.,  we  find  that  they  are 
all  on  a  certain  straight  line. 

Let  US  investigate  this  question  in  general.  All  of  those 
values  of  a;  and  //,  which  cause  the  function  ax  -f-  by  +  c  to 
assume  a  given  value  h,  must  satisfy  the  equation 

(1)  ax  -\-by  +  <■  =  k,  or 

(2)  .  ax+by  +  c  -k  =  0. 

But  we  have  seen  in  Art.  54  that  all  of  those  points 
whose  coordinates,  x  and  y,  satisfy  an  equation  of  the  first 
degree  are  on  a  straight  line.  Similarly  those  values  of  x 
and  y  which  cause  the  function  ax  -{-by  +  c  to  assume  a  sec- 
ond given  value  k'  correspond  to  the  points  of  a  second 
straight  line,  namely  to  the  locus  of  the  equation 

(3)  ax  +  by  +  c  -k'  =  0. 

If  6  =  0,  the  lines  (2)  and  (o)  are  both  parallel  to  tiie  y-axis 
(see  Art.  54).  \{  b  is  not  equal  to  zero,  both  lines  have  the 
same  slope,  namely,  —a/b  (see  Art.  54).  Thus  the  two 
lines  (2)  and   (3)  are  parallel  in  either  case. 


AuT.  1!)9]  LINEAR  EQUATIONS  321 

We  hiive  found  the  following  resnlt.  //  we  interpret  x 
and  y  as  the  rectanr/ular  coordinates  of  a  point  in  a  plane,  all 
of  those  points  whose  coordinates  cause  a  linear  function,  such 
as  a.v  +  l>t/  -\-  c.  to  assume  a  given  value  k  are  on  a  sfraic/ht  line. 
Thus  there  is  one  such  straight  line  for  everg  value  of  k,  and 
the  straight  lines  wli/rh  correspond  to  any  two  dttferent  values 
of  k  are  parallel. 

EXERCISE    LXXXVI 

1.  Plot  the  straight  line  upon  whicli  the  function  f{x,  1/)=  x  +  1/  +  I 
assumes  the  value  —  ^3 ;  also  jilot  the  lines  upon  which  f{x,y)  =  —  2 
f{x,  //)--!,  /(x,  y)  =0,  f{x,  y)  =  +  1,  /(x,  ^)  =  +  12,  /(x,  y)=  +  -d. 
Observe  that  the  straight  line  upon  which  f{x,  y)  =  0  divides  tlie  plane 
into  two  portions  such  that  ./'(x,  y)  is  negative  on  one  side  of  this  line 
and  positive  on  the  other. 

2.  Study  each  of  the  following  functions  by  the  nietiiod  outlined  in 
Example  1. 

(a)    2x-^  +  3.  (c)    3a:-4.y+5.  (e)    2x  +  ^y-l. 

\b)    x  +  2y  +  3.  {<!)    4x  +  3y  +  5.  (/)    3x-2y  +  2. 

199.  Linear  equations.  We  have  seen  in  Art.  198  that 
those  values  of  x  and  y  ivhich  cause  the  linear  function  ax  +  by 
■+■  c  to  assume  the  value  k,  are  exactly  the  same  as  those  which 
cause  another  linear  function,  namely  ax  +  by  +  c  —  k,  to  assume 
the  value  zero. 

The  following  problem  is  therefore  especially  important. 
Given  any  linear  function  ax  +  5y  +  c.  To  find  the  values 
of  X  and  y  which  cause  it  to  assume  the  value  zero;  or,  in 
other  words,  to  find  all  of  those  values  of  x  and  y  which 
satisfy  the  equation 
(1)  ax  ^by  -\-c  =  (^. 

An  equation  of  this  form,  obtained  by  equating  to  zero  a 
linear  function  of  two  variables,  is  called  a  linear  equation 
or  an  equation  of  the  first  degree  with  two  unknowns.  Any 
pair  of  numbers,  x  and  ;y,  which  satisfies  such  an  equation 
is  called  a  solution  of  (1  ). 

Observe  that  it  takes  an  ./■  and  a  //.  tliat  is,  a  pair  of  numbers,  to  con- 
stitute a  solution.  Thus  the  equation  x  +  //  —  .5  =  0  has  as  one  of  its 
solutions  X  =  1,  ^  =  4.     Other  solutions  are  x  =  0,  y  =  o ;  x  =  2.  ?/  =  .3. 


322     LINEAR   EQUATIONS  AND  DETERMINANTS      [Art.  200 

Every  equation  of  the  form  (1)  has  infinitely  many  solu- 
tions. These  may  be  obtained  as  follows.  If  ^  ^t  0  we  find 
from  (1),  by  solving  for  y, 

(2)  ^  =  -1^-1- 

If  we  give  to  x  any  value  whatever,  and  use  (2)  to  compute 
for  every  x  a  corresponding  value  of  y,  the  pairs  of  corre- 
sponding numbers,  x  and  «/,  obtained  in  this  way  will  all  be 
solutions  of  (1)*. 

If,  however,  ft  =  0,  (1)  reduces  to  ax  ■}-  c  =  0  and  this  gives 

(3)  x  =  --.  ' 

a 

wliich  determines  x  completely,  but  allows  y  to  have  any 
value  whatever.  In  this  case  also  we  have  infinitely  many 
solutions. 

We  first  considered  linear  equations  in  Art.  54.  We 
proved  there  that  such  an  equation  defines  y  as  a  linear 
function  of  a:  if  b  is  different  from  zero,  and  that  in  all  cases 
the  graph  of  an  equation  of  the  first  degree  between  x  and  y 
is  a  straight  line.  This,  in  fact,  was  the  reason  for  speak- 
ing of  such  equations  as  linear  equations. 

EXERCISE  LXXXVII 

From  each  of  the  following  equations  express  y  as  a  function  of  x,  or  x 
as  a  function  of  y,  whenever  possible,  and  draw  the  corresponding  line. 

1.  2x  +  y  =  1.  4.   2x4-8  =  0. 

2.  3  X  -  4  //  =  .5.  5.    Sy  -  9  =  0. 

3.  4x-  7  y  =  0.  6-    |  +  f  =  1- 

200.  Simultaneous  linear  equations.  We  have  just  seen 
that  there  are   infinitely  many  pairs  of  values  of  x  and  y 

*  For  let  X  have  any  value,  and  let  ?/  =  —     ^  — ,  •    Then 


ax  +  by  +  c  =  ax  +  b  {—  -  X  —  -\  + 


c  =  ax  —  ax  —  c  +  c  =  0. 


That  is,  every  pair  of  numbers  obtained  in  this  way  will  actually  satisfy  the 
given  equation. 


Akt.  200]      SIMULTANEOUS  LINEAR   EQUATIONS  323 

which  satisfy  a  single  equation  of  the  form 

(1)  a^x  +  h^y  +  (?j  =  0, 

and  that  each  of  these  pairs  may  be  interpreted  as  the  co- 
ordinates of  a  point  on  a  certain  straight  line,  the  locus  of 
the  equation.     Similarly,  the  solutions  of  a  second  equation 

(2)  a^x  +  h.^y  +  c.^  =0,* 

correspond  to  the  points  of  a  second  straight  line.  If  these 
two  lines  are  not  parallel  and  if  they  do  not  coincide,  they 
will  have  just  one  point  of  intersection.  The  coordinates  of 
the  point  of  intersection  must  satisfy  each  of  the  two  equa- 
tions (1)  and  (2),  and  it  is  the  only  point  whose  coordinates 
satisfy  both  of  these  equations. 

The  coordinates  of  the  pointy  in  which  the  lines  represented 
hy  the  two  equations  intersect^  may  therefore  be  obtained  hy 
solving  the  equations  (1)  and  {'2)  simidtaneously^  that  is^  by 
finding  the  values  of  x  and  y  which  will  satisfy  both  of  these 
equations  at  the  same  time. 

This  statement  shows  us  how  to  solve  a  certain  problem 
of  geometry  (intersection  of  two  lines)  by  algebra  (solu- 
tion of  equations. (1)  and  (2)).  We  may  just  as  well,  how- 
ever, regard  this  theorem  as  the  geometric  solution  of  an 
algebraic  problem.     For  it  may  be  restated  as  follows : 

To  solve  the  equations  (1)  and  (2)  simultaneously.,  we  draw 
the  lines  which  are  the  loci  of  (1)  and  (2)  and  deternmie  the 
coordinates  of  the  point  of  intersection  by  measurement. 

However,  it  should  be  remarked  that  in  general  this 
later  method  will  give  us  only  an  approodmate  value  of 
the  solution. 

*  The  notation  oj,  uo,  h^,  bo,  etc.,  is  very  convenient.  As  liere  used  the  letter 
a  indicates  a  coefficient  of  x;  Uy  for  the  first,  a-i  for  the  second  equation.  Sim- 
ilarly ^1  denotes  the  coefficient  of  y  in  equation  (1),  and  bo  the  coefficient  of  i/ 
in  (2). 


324     LINEAR  EQUATIONS  AND  DETERMINANTS      [Art.  201 


EXERCISE   LXXXVIII 

Solve   the   following    pairs    of    equations    both    arithmetically   and 
graphically. 


1. 


2. 


3. 


4. 


/  x  4-  ?/  =  2, 
\x  -  y  r=2. 
j2x  +  y=  1, 
\x-2>/  =  0. 

I  3  X  —  I  //  =  5, 


6     /  3  X  =  I/, 

\l/  -6x  +  2  =  0. 
J      j  X  +  my  —  1, 

\y  =  mx. 


^^  +  y 


8. 


1, 


f  -  ^  =  1 
I  ft      a 

^  +  f  =  l, 
a      0 


Hint.  In  plotting, 
[4  x  +  o  y  =  5.  regard  in  as  a  positive 
J  X  +  .y  —  3  =  0,       number,  and  use  a  line-  1  -^  ~  "*"^' 

\x  —  y  +  i  =  0.       segment   of    convenient  length  to  represent  m. 
f  X  +  fj  —  'd  =  0      Similarly  for  a,  b,  and  m  in  Examples  8  and  9. 
\y  :^7  X. 


201.  General  formulas  for  the  solution  of  two  simultaneous 
linear  equations  with  two  unknowns.  Let  us  transpose  the 
terms  c^  and  c^  of  equations  (1)  and  (2)  of  Art.  200,  and  let 
us  call  —  <?j  and  —  c^,  k^  and  k,^  respectively,  so  that  the 
equations  assume  the  form 


(1) 
(2) 


a^x  -f  h^  =  k^' 


To  solve  these  equations  we  first  eliminate  y.  This  may  be 
done  by  multiplying  both  members  of  (1)  by  h^,  and  both 
members  of  (2)  by  —  h^,  ;ind  adding.      We  find 

Similarly,  elimination  of  x  from  (1)  and  (2)  gives 

(4)  (^/'2  ~  "Jh  \'l  ~  "i^-i  ~  "•'^"r 

If  a^b^  —  (i.y^'i  ^^  'i^t;  e(jual   to  zero  wc  find,  from   (3)  and 
(4),  the  values 


(5) 


k.b^  —  k.,b,  a^kc,  —  aJc-, 


and  it  is  easy  to  verify,  by  actual  substitution,  that  these 
values  of  x  and  y  really  satisfy  both  equations  (1)  and  (2). 


Akt. -'(iL']     DETERMINANTS  OF   THE   SECOND   ORDER        325 

Thus,  the  solution  of  the  simultaneous  equations  {\)  and  (2) 
is  given  hy  (5),  provided  that  a^h^  —  a^^  is  different  from 
zero;   and  there  exists  only  one  solution  in  this  case. 

202.  Determinants  of  the  second  order.  The  values  of  x 
ami  y  as  i^iveu  by  (5),  Art.  201,  appear  as  fractions  with  a 
euuiinon  denominator.  This  denominator  is  made  up  en- 
tirely of  the  four  numbers  a^,  b^,  a^,  b^,  which  appear  in 
equations  (1)  and  (2)  of  Art.  201  as  coefficients  of  x  and  y. 
Let  us  write  these  coefficients  in  the  order  in  which  they 
appear  in  these  equations,  thus  : 


and  let  us  introduce  a  new  symbol  made  up  of  these  four 
numl)ers  written  between  two  vertical  lines,  thus  : 


/>, 


a., 


and  let  us  define  this  symbol  by  saying  that  it  shall  be  equal 
to  aj).,  —  ac,b^^  tlie  common  denominator  of  the  two  fractions 
(5),  Art.  201. 

We  have  then  by  detinition 


(1) 


=  aj>.,  —  a^by 


This  quantity,  a  siiiirlc  number  fui'nied  fi-oin  the  f(jur  num- 
bers flj,  ^p  (^2'  ^^T  '^^  indicated  in  (1),  is  called  a  determinant 
of  the  second  order. 


Tims,  we  hiive 


o     -1 
3     2 


_  1  .  ;i  =  10  _  12 


We  now  observe  tliat  the  nuniei'ators  of  the  expressions 
(o)  of  Art.  201  may  also  be  written  as  determinants,  namely: 

Ir  I 

/rj* 


CLyfCty    CLk^fC't     


"1 
a., 


326      LINEAR  EQUATIONS  AND  DETERMINANTS      [Art.  203 

We  have  therefore  the  following  new  form  for  the  solution 
of  equations  (1)  and  (2)  of  Art.  201  : 


(2) 


h 

h 

«i 

^1 

^2 

h 

'  I/  = 

«2 

h 

«1 

h 

«1 

h 

«2 

K 

^2 

h 

Observe  that  the  denominator  of  both  expressions  is  the 
same  determinant,  namely,  that  one  which  is  formed  from 
the  coefficients  of  x  and  i/  in  the  equations 

a^x  +  h^y  =  k^. 

Observe  further  that  the  numerator  of  a::  is  a  determinant  ob- 
tained from  the  denominator  determinant  by  replacing  in  it 
the  coefficients  of  x  (a^  and  a^)  by  the  right  members  (^k^^ 
and  k^').  Observe  finally  that  the  expression  in  the  nu- 
merator of  9/  is  a  determinant  obtained  from  the  denominator 
determinant  by  replacing  in  it  the  coefficients  of  t/  (b-^  and  Jj) 
by  the  right  members  {k^  and  ^2)- 


EXERCISE    LXXXIX 

Find  the  values  of  the  foUowino-  determinants: 


1. 


2. 


I  2 
3     4 

II  4 
2     5 


3. 


1     -  6 

2-3 

-26 

2     5 


a     b 

c     d 

-d 

h 

c 

—  a 

7-15.    Solve  the    equations    given    in    Examples    1-9    of    Exercise 
LXXXVIII  by  using  determinants. 

203.   Homogeneous  linear  equations  with  two  unknowns.     If 

the  right  members,  k^  and  k^^  are  both   equal  to  zero,  the 
ec^uations  are  said  to  be  homogeneous. 
Tlius,  the  two  equations 


(1) 
(2) 


a^x  -f  h^y  =  0, 
a^x  +  h^y  =  0. 


Ai:t.  2o:5]       HOMOGENEOUS   LINEAR   EQUATIONS  827 

form  a  Jioniogeneous  system.     If  the  determinant 

is  different  from  zero,  equations  (2)  of  Art.  202  show  that 
the  only  solution  of  (1)  and  (2)  is  .'r  =  ?/  =  0,  since  both  of 
the  numerators  which  occur  in  (2),  Art.  202,  will  vanish 
when  Tc^  and  h.^  are  equal  to  zero.  The  solution  ./•  =  y  =  0  of 
the  equations  (1)  and  (2)  is  often  called  an  obvious  or  trivial 
solution,  because  it  is  evident  that  every  system  of  this  form 
has  this  solution. 

But  if  the  determinant  D  is  equal  to  zero,  it  does  not  follow 
that  the  system  (1)  (2)  has  x=i/  =  0  as  its  only  solution, 
since  the  expressions  (2)  of  Art.  202  become  useless  in  this 
case,  on  account  of  their  indeterminateness. 

We  can  easily  show  that  in  this  case  (when  D  =  0),  there 
do  exist  other  solutions  of  (1)  and  (2),  besides  the  obvious 
one  x=  7/  =  0.     In  fact,  if  we  put  in  (1 ) 

(4)  X  =  kh^,   y  =  —  ka^, 

where  Jc  is  any  number  whatever,  equation  (1)  will  be  satis- 
fied siuce,  for  these  values  of  x  and  ij,  we  find 

a^r  -\-  h-^ij  =  a^kh^  +  /)^(  —  ka^)  =  ki^a^h^—  a^h^  =  0. 

But  these  same  values  of  x  and  y  will  also  satisfy  (2);  for 
we  have 

a,2^-  +  h^jj  =  a.Jch^  -\-  h.^(  —  ka^')  =  —  k(a^b^  —  a^h^)  =  0, 

since,  in  our  case,  D  =  a-jb.,  —  a.yb^  is  equal  to  zero.  Thus  all 
of  the  pairs  of  numbers,  which  can  be  obtained  from  (4)  by 
giving  different  values  to  k,  will  be  solutions  of  both  (1)  and 
(2).  Moreover,  all  of  these  solutions  will  not  coincide  with 
the  obvious  solution  x=  y  =  0  unless  a^  and  5j  are  both  equal 
to  zero.  Let  us  exclude  this  case  and  also  the  analogous  case 
in  which  a^  and  b^  are  both  equal  to  zero,  since  in  either  of 
these  cases  we  should  really  be  dealing  with  only  a  single 
equation ;  one  at  least  of  the  equations  (1)  or  (2)  might 
then  be  resrarded  as  absent. 


328     LINEAR  EQUATIONS  AND  DETERMINANTS      [Art.  2():J 

We  may  summarize  our  results  as  follows: 

If  the  determinant  D  of  the  homogeneous  equations  (1)  and 
(  2 )  ?'.s'  not  equal  to  zero,  these  equations  have  only  a  single  solu- 
tion, namely,  the  obvious  one,  x  =  y  =  Q.  If  T)  is  equal  to  zero, 
the  two  equations  have  infinitely  many  solutions.  More  specifi- 
cally we  can  say  that  if  both  equations  are  actually  present,  that 
is,  if  neither  of  them  has  both  of  its  coefficients  equal  to  zero,  then 
if  D  is  equal  to  zero,  the  two  equations  are  equivalent ;  that  is, 
every  solutio7i  of  one  equation  is  also  a  solution  of  the  other. 

Although  the  equations  (1)  and  (2)  are  satisfied  by  infi- 
nitely many  pairs  of  values,  x  and  y,  \i  D  =  0,  the  ratio  of  x 
to  y  will  nevertheless  be  determined  uniquely.  In  fact,  this 
ratio  may  be  computed  from  each  of  the  two  equations,  and 
the  resulting  two  values  will  be  found  equal  on  account  of 
the  relation  2)  =  0. 

Thus,  the  homogeneous  equations  (1)  and  (2)  forx  and  y  may 
be  regarded  as  two  no7i-homogeneous  equations  for  a  single  un- 
known, the  ratio  of  x  to  y  ;  the  condition  that  these  two  equations 
7nay  be  consistent,  that  is,  that  they  shall  furnish  the  same  value 
for  this  ratio,  is  again  D  =  0. 

To  clarify  still  more  the  significance  of  the  condition 
i>=  0,  Ave  add  the  following  remarks. 

If  a^  and  b^  are  common  multiples  of  a^  and  b-^,  that  is,  if 

a^=  ma-^,    b^  =  mb^, 

the  determinant  D  =  a^b^  —  agi^  is  equal  to  zero.     Conversely, 
if  i>  =  0  and  if  a^  and  ^j  are  not  equal  to  zero,  we  have 

a  A  -  ^A  =  0,  or  h  =  ^. 

If  we  denote  by  m  tlie  common  value  of  tliese  two  fractions, 

we  find 

«2  =  ma-y,   ho  =  mh-^, 

so  that 

(5)  a^x  +  />2,y  =  m(ayr  +  ftj?/). 


Art.  204]  DISCUSSION   OF   GENERAL   CASE  329 

Therefore,  if  the  determinant  D  of  (1)  and  {2)  is  equal  to 
zero,  and  if  not  both  of  the  coefficients  of  either  equation  are 
equal  to  zero,  then  the  coefficients  of  one  of  the  equations  will  he 
proportional  to  those  of  the  other  ;  in  other  ivords,  the  left  mem- 
ber of  one  of  these  equations  will  be  a  mere  multiple  of  the  left 
member  of  the  other. 

Ill  our  proof  we  have  assumed  that  Wj  and  i,  are  liotli  different  from 
zero.  The  student  should  complete  the  proof  by  considering  separately 
the  cases  a-y4^(),  6,  =  0,  and  r/j  =  0,  b^  =^  0,  the  further  case  rr,  =  />,  =  0 
being  excluded  by  our  liypothesis  that  not  both  of  the  cofficientsof  either 
ecpiation  shall  vanish.  The  student  may  easily  decide,  however,  what 
would  happen  in  this  case  also. 

The  theorems  of  tliis  article  are  easily  explained  graphi- 
cally. The  graplis  of  the  two  e(iiuiti()ns  (1)  and  (2)  are 
straight  lines  through  the  origin  of  coordinates.  If  the  deter- 
minant D  is  not  equal  to  zero,  these  lines  are  distinct  and 
their  only  common  point  is  the  origin,  x=  y  =  0.  lfi)  =  0, 
the  two  lines  coincide,  and  have  all  of  their  points  in  common. 

204.  Discussion  of  the  solutions  of  two  linear  equations  with 
two  unknowns.      We  have  seen  tliat  the  equations 

(1)  a^T  +  b^i/  =k^, 

(2)  a.^.r  +  k,!j  =  k^^, 

have  a  single  solution  if  i>=^0.  (See  Arts.  201  and  202.) 
If  2>=  0,  we  tind  from  (3)  and  (4)  of  Art.  201, 

(3)  k^b.,  -  kj^^  =  0, 

(4)  a^k^  —  ajc^  =  0, 

so  that  k^  and  k.y  must  satisfy  these  conditions  if  (1)  and  (2) 
are  to  have  any  solutions  at  all.  Bnt  the  left  members  of 
(3)  and  (4)  are  the  determinants  which  occur  in  the  numer- 
ators of  the  expressions  (2)  of  Art.  202  for  x  and  /y. 

Thei'efore,  the  si/stem  of  equations  (1)  and  (2)  has  no  solu- 
tion at  all  if  D  =  U.  unless  at  the  same  time  both  of  the  determi- 
nants 


330     LINEAR  EQUATIONS  AND  DETERMINANTS      [Art.  204 


(5) 

1    '*i 
and 

l«2 

k, 
k., 

are  equal  to  zero. 

If 

(6)                 i>=0, 

=  0,  and 

k, 
k^ 

=  0, 


then  (1)  awe?  (2)  have  i7ifimtel'y  many  solutions.  Unless  one 
of  the  two  equations  has  all  of  its  coefficients  equal  to  zero.,  each 
of  the  two  equations  may  he  obtained  from  the  other  by  multiply- 
ing both  of  its  members  by  an  appropriately  chosen  number.,  and 
the  tivo  equations  are  equivalent,  that  is.,  they  have  exactly  the 
same  solutions. 

In  fact  the  three  equations  (6)  are  equivalent  to  the  con- 
tinued proportion 

a-^  :  b^  :  k^  =  a^  '•  bo  :  ko- 

If  we  remember  that  each  of  the  equations  (1)  and  (2)  has 
for  its  graph  a  straight  line,  we  see  at  once  that  the  corre- 
sponding geometrical  situation  may  be  described  as  follows  : 

The  graphs  of  equations  (1)  arid  (2)  are  distinct  intersect- 
ing lines  if  D  4^  0.  They  are  distinct  parallel  lines  if  D  =  0 
and  if  not  both  of  the  determinants  (5)  are  equal  to  zero.  The 
two  graphs  coincide  throughout  if  conditions  (6)  are  satisfied. 

EXERCISE     XC 


First  discuss  the  following  pairs  of  equations  by  the  methods  of  Arts. 
202  and  204.     Then  find  all  of  their  solutions. 


1. 


/  2  X  +  .V  =  1, 


'■^y 


/  X  +  ?/  =  5, 
\2x  +  2y  =  1. 

j  X  +  y  =  o, 
\2x  +  2y=10. 


4    f:3.r--ry-.5  =  0. 
■  \  2  X  +  3  y  +  1  =  0. 


5    {3,r-4.v-7  =  0. 
■  \l8x-24?/  =  38. 


6    f2x+3?/-7  =  0. 
■  \8a;  +  12y  =  28. 


7_  /  y  =  mx  +  h, 

\y  =  m'x  +  h', 

m  ^  m'. 

Q    \  y  =  mx  +  h, 

\y  =  mx  +  b', 

h  ^  h'. 


a      0 


1, 


y  =  h--x. 
a 


Art.  205]     DETERMINANTS   OF   THE   SECOND   ORDER        331 

10.    ^Vl^at  value  or  values  must  /:  have  in  order  that  the  equations 
3  X  +  4  y  =  0, 

1-2  X  +  khj  =  0, 

may  have  other  solutions  besides  x  =  ?/  =  0?     Find  these  solutions  when 
thev  exist. 


205.   Properties  of  determinants  of  the  second  order.     The 

following  theorems  are  important  for  what  follows  : 

1.  A  determinant  of  the  second  order  does  not  change  its 
imlue  if  its  columns  {vertical  lines^  are  converted  into  rows 
{horizontal  lines^^  that  is. 


a.     a 


b^     h 


Proof.     Both  members  are  equal  to  aih^  —  a-ihi. 

2.    A  second  order  determinant  is  equal  to  zero  if  both  ele- 
ments of  any  one  of  its  rows  or  columns  are  equal  to  zero. 


For  instance 


(I.,     b.j 


0.     if  Wj  =  a„  =  0. 


3.    A  determinant  of  the  second  order  chanc/es  its  si<jn  if 
either  its  columns  are  interchanged  or  its  roivs  are  interchanged. 


aj)^  —  a^b.2, 


Proof.     We  1 

lave 

D  = 

=  a  ^1.2 

—  (i„h^, 

b,     a, 
h.,     a., 

and  therefore 

])'  =  -  D. 

4.  If  the  elements  of  any  column  {or  roiv)  have  a  common 
factor,  this  factor  may  be  removed  from  that  column  {or  row') 
and  placed  in  front  of  the  whole  determinant. 


Proof.     We  have 


maxho  —  maih  =  m(a\h-2  —  o-2?n)  =  m 


k'l     bi 
ao     b', 


332     LINEAR  EQUATIONS  AND  DETERMINANTS      [Aht.  liUG 

5.  A  deternunant  of  the  second  order  is  equal  to  zero  if  the 
corresponding  elements  in  two  parallel  columns  (^or  rozvs)  are 
equal  or  proportional. 

ri:ooji\     If  hi :  b-2  =  ui :  a.^,  we  may  put  bi  =  ma\,  h>  —  uki-^,  and  then 
ai     In 


a.2     />■ 


=  (iih-2  —  (t-Jn  =  m(^a\(i-2  —  </!»•_>)  =  0. 


6.  If  evert/  eleynent  of  any  column  (or  roiv^  is  expressed  as  a 
sum  of  two  terms,  the  determinant  may  he  rewritten  as  a  sum 
of  two  determinants  as  in  the  folloiving  cases  : 


(1) 


«1 

+  ^1, 

h 

«i 

^1 

^1 

h 

^h 

+  ^2' 

h 

«2 

h 

+ 

^2 

h 

a^. 

h  + 

d. 

«1 

h 

+ 

«1 

d. 

<h^ 

h  + 

d^ 

«2 

K 

a^ 

d^ 

(2) 


Proof.  Expand  both  members  of  (1)  and  compare  the  results. 
>^imilarly,  for  (2). 

7.  The  value  of  a  second  order  determinant  is  not  altered  if 
to  the  elements  of  any  colunm  (or  row^  he  added  common  multi- 
ples of  rorrcsponding  elements  of  a  parallel  column  (or  roiv'). 

Proof.     According  to  theorem  (J  of  this  article,  we  liave 


Oi  +  mbi, 

bi 

<h 

hi 

■m1)\ 

-L  1 

bi 

02  +  mb'i, 

h 

«2 

1-2 

1  nih'i 

h-2 

])ut  the  second  determinant  in  the  riglit-hand  member  is  equal  to  zero  on 
account  of  theoreni  ,5. 


206.    Determinants  of  the  third  order.      Let  it  be  required 
1(»  solve  the  L'(|uuti(>n.s 


(1) 

(2) 
(3) 


a^x  +  b^y  +  c^z  =  r^, 
a^x  +  h^y  +  c^z  =  /"a, 
a^x  +  h^f/  +  c^z  =  rg, 


for  X,  y,  and  z.      We  proceed  just  as  thongli  the  coefficients 
were  specilic  given  numbers,  and  eliminate  y  and  z,  so  as  to 


Art.  206]     DETERMINANTS   OF   THE    TIIIKl)   OllDER 


333 


obtain  an  equation  invulving-  x  alone.     To  accomplish  this 
we  rewrite  (2)  and  (3)  as  follows: 


(4) 


hi/  +  ''2~  =  ''2  -  "2^' 
^3^  +  ^'3-  =   >'-6  -  «3^' 


and  solve  (4)  for  ?/  and  z.     This  gives,  according   to  (2) 
Art.  202,  and  theorem  6  of  Art.  205, 


(5) 


y  = 


z  = 


a^x,     c^ 

_    '"2 

e^ 

|_ 

^2 

^2 

(f^X,        6'3 

'•3 

^3!         I«3 

^3 

^2  -  «2^ 

^2 

''2 

^2 

^2 

^3  -  «3^' 

h 

^'3 

*3 

«3 

X, 


where  we  have  not  divided  by  the  coefficient  of  ^  and  z,  thus 
avoiding  cumbersome  fractions. 

Let  us  substitute  the  values  of  y  and  z  from  (5)  into  (1). 
But  again,  so  as  to  avoid  fractions,  let  us  first  multiply  both 
members  of  (1)  by 

f>2     ^2 


We  shall  then  find,  after  substituting  and  uniting  all  of  the 
terms  which  contain  a:  as  a  factor, 


^'2     ^'2 
^'3     ^3 


+  h 


2 


+   C, 


h     ^'3 


=  r 


'\h     <'. 


or  (making  use  of  Theorem  3  of  Art.  205) 


(6) 


^2       ^^2 


I  ^2       ^2 
1  *3       H 


-h. 


-h. 


«2       H 

/  1}  Oct 


+  ^1 


+  ^1 


«2        f'2 

^2     ^2! 


The  coefficient  of   x,  in  this  equation,  depends  upon  the 
nine  coefficients  aj,  6j,  e^,  a^^  \^  c^,  a^,  63,  Cg,  of  the  three  un- 


334     LINEAR  EQUATIONS  AND  DETERMINANTS      [Art.  206 


^1 

^2 

=  «1 

^2 

^3 

«3 

^2 
^3 

+  ^1 

«2 

^2 
^'3 

^3 

kiiowiis  in  the  three  equations  (1),  (2),  (3).  It  does  not 
depend  upon  the  values  of  r^,  r^^  r^  We  shall  henceforth 
speak  of  this  coefficient  as  the  determinant  of  these  nine 
quantities,  and  use  the  symbol 

h 

to  represent  it.     Thus  we  are  led  to  the  following  definition  : 

When  nine  Clumbers  are  writteji  in  the  form  of  a  square  array, 
the  value  of  their  determinant,  represented  hy  the  symbol  (7), 
is  hy  definition  equal  to  the  coefficient  of  x  in  (6). 

We  have  therefore  the  following  defining  equation  for  a 
determinant  of  the  third  order : 


(8)     D  = 


This  formula,  whicli  we  have  used  as  a  definition  for  the 
symbol  in  the  left  member,  is  easy  to  remember.  Each  of 
the  three  terms  of  the  right  member  is  a  product  of  two 
factors ;  one  of  the  two  factors  of  each  of  these  terms  is  one 
of  the  elements  a^,  6j,  c^  of  the  first  row  of  the  determinant ; 
the  other  factor  is  that  second  order  determinant  which  we 
obtain  from  the  tliird  order  determinant  (8)  by  crossing 
out  the  row  and  the  column  in  which  stands  that  one  of  the 
elements  rtp  h^,  c^,  wliich  is  tlie  first  factor  of  the  term  in 
question.      The  signs  of  the  three  terms  are  in  order  +,  —  •>  +• 

Let  us  substitute,  in  (8),  for  the  second  order  determi- 
nants their  values  and  multiply  out.      We  find  in  this  way 

B  =  a^(h^c^  -  h^c^)  -  ^'lO'gC'g  -  a^c^  +  c^^a^h^  -  a^h^), 

which  may  be  rewritten  as  follows ; 

and  this  expression  might  also  be  used  as  a  definition  for  the 
determinant  D.     We  see  that   a    determinant  of   the  third 


Art.  206]     DETERMINANTS   OF   THE   THIRL)   ORDER 


335 


order  consists  of  6  terms,  each  of  wliich  is  a  product  of  three 
of  the  nine  numbers  a^,  ■••  Cg,  three  of  tliese  products  being 
preceded  by  the  plus  sign,  and  the  other  three  b}^  the  minus 
sign. 

We  find  again  the  expression  (8)  for  D  from  (9),  if  we  unite  tlie 
two  terms  of  (!))  which  contain  a,,  the  two  terms  which  contain  ftj,  and 
tlie  two  terms  which  contain  Cy  Let  us  instead  unite  the  terms  which 
contain  a^,  b^,  cv-     Then  we  may  write  D  as  follows: 


(10) 


D 


aj'^ 

^1 

+  h 

«i 

^1 

—   C.y 

"\ 

h 

'\h 

^3 

^'3 

'•3 

«3 

h 

a  new  expression  for  D  similar  to  (8).  There  are  again  three  terms. 
But  this  time  the  first  factor  of  each  term  is  an  element  of  the  second 
row  cr„,  h„,  c„.  The  second  factor  of  each  term  is  again  obtained,  as  in 
(8),  as  a  second  order  determinant  by  suppressing  that  row  and  column 
of  D  to  which  the  first  factor  of  the  term  belongs. 

In  order  to  express  this  hiw  of  formation  more  compactly 
we  introduce  a  new  word  by  the  following  definition. 

The  minor  of  a  particular  one  of  the  nine  eleynents  of  the  third 
order  determinant  ,         , 


D  = 


h. 


is  that  second  order  determinant  which  is  ohtained  from  D  if 
the  row  and  column  he  erased  to  which  that  particular  elernent 
of  D  belongs. 

Thus  the  minors  of  n^,  h^,  c^  are  respectively 


*2 

^2 

) 

«1 

^1 

"1 

h 

h 

<^3 

"3 

<^3 

a„ 

b.^ 

We  can  now  say  that  the  two  expressions.  (8)  and  (10),  for  D  have 
this  in  common.  Each  term  of  either  expression  is  the  product  of  an 
element  of  D  by  the  minor  of  that  element.  The  two  expressions  differ 
in  so  far  as  the  elements  which  are  used  in  (8)  are  those  of  tlie  first  row, 
while  those  used  in  (10)  belong  to  the  second  row.     Moreover,  the  terms 


336      LINEAR  EQUATIONS  AND  DETERMINANTS      [Art.  206 

are  preceded  by  the  signs  +,  — ,  +  in  (8),  and  by  the  signs  — ,  +,  —  in 
(10). 

In  order  to  get  to  the  bottom  of  this  matter,  we  now  arrange  the 
expression  (9)  for  D  according  to  the  elements  a^,  b.^,  c^  of  the  third  row. 
We  find 


(11) 


D  =  cu 


-K 


a„     c, 


+  c^ 


"i    h 
a„     h. 


Tlie  resnlts  (8),  (10),  and  (11)  may  be  summed  up  as  follows  : 

We  may  expand  a  determinant  D  of  the  third  order  accord- 
in;/  to  the  elements  of  the  kth  row^  where  k  may  he  equal  to  1,  2, 
or  3.  This  expansion  vnll  present  D  as  a  sum  of  three  prod- 
ucts of  the  form 

1st  element  of  the  kth  row  times  its  minor, 
2d  element  of  the  kth  roiv  times  its  minor, 
?>d  element  of  the  kth  row  times  its  minor. 

Each  of  these  products  is  preceded  by  a  plus  or  nmius  sign, 
which  is  determined  in  accordance  with  the  following  diagram 
of  signs : 

(12)  ^=  _      +      - 

+      -      + 

Tliree  further  expressions  for  I)  may  he  obtained  by  expand- 
ing with  respect  to  the  elements  of  any  column  according  to  the 
same  rule,  using  the  columns  in  the  diagram  of  signs. 

To  prove  tlie  last  statement,  it  suffices  to  arrange  D,  as 
given  by  (9),  with  respect  to  the  elements  a^,  a^,  a^  of  the' 
first  column,  or  with  respect  to  b^  b^,  b^,  or  c^  c^,  Cg. 


EXERCISE  XCI 

Compute  the  values  of  the  following  determinants. 


1      1      1 

11 

1 

1 

;3    4 

2     -  6      -  1 

2. 

0 

-6 

-  1 

3. 

7    :} 

3          1          2 

0 

4 

2 

:5    .5 

:i         2 

:} 

2     1 

1         1 

-  1     -  1     -  :} 

5. 

■1 

-  (i     2 

6. 

3         0 

2         4     - 

1 

1 

0     1 

0 

1     - 

.0 

207] 

COFACTORS 

«     0     0 

a     h     g 

a     h     g 

0     b     0 

8. 

h     b   / 

9. 

a     k     g 

0     0c 

9    f    c 

9     f    <^ 

337 


7. 


207.  Cofactors.  The  six  expansions  of  a  tliiid  order  de- 
terminant wliit'h  we  have  found  in  Art.  20(3  assume  a  some- 
what simpler  form  if  v,e  introduce  tlie  notion  of  cofactor  in 
place  of  the  notion  minor.  Every  element  of  the  determi- 
nant D  has  associated  with  it,  as  its  minor,  a  certain  detei- 
minant  of  the  second  order.  The  expansion  of  D  with 
respect  to  the  elements  of  one  of  its  rows  or  columns  is  com- 
plicated by  the  fact  that  some  of  the  terms  of  such  an 
expansion  are  preceded  by  the  minus  sign.  We  may  get  rid 
of  this  complication  by  uniting  the  minus  sign  with  the 
corresponding  minor,  in  accordance  with  the  following 
definition. 

Consider  the  third  order  determinant  D  and  the  diagram  of 
signs  S,  defined  by 


D 


K 


S= 


+ 

— 

+ 

— 

+ 

— 

+ 

- 

+ 

£g  the  cofactor  of  a  given  element  of  Z>,  we  mean  the  minor 
of  that  element  preceded  by  the  plus  or  minus  sign,  according 
as  the  place  in  the  diagram  S  ivhich  corresponds  to  the  given 
element  of  D  is  occupied  by  a  plus  or  minus  sign. 

Thus,  the  cofactor  and  the  minor  of  a  given  element  of  D  differ  at 
most  in  sign.     The  cofactor  of  c,  is 

If  the  cofactors  of  a^,  ftj,  Cj,  etc.,  be  denoted  by  A^,  B^,  C\, 
etc.,  tve  may  noiv  write  Din  any  one  of  the  following  sir  forms. • 

D  =  a^A^  4-  b^B^  +  c^  C^,     D  =  a^A^  -f  a^A^_^  4-  'a^Ay. 

(1  )    B  =  a^A^  -f-  ^2^2  +  ^I^V       ^  =  ^^1  +   ^2^2  +  ^'3^3' 

B  =  a^A^  +  b^B^  +  t-gCg,     B  =  c^C^  +  c.,t\  +  c^C^ 


338     LINEAR  EQUATIONS  AND  DETERMINANTS      [Art.  208 


EXERCISE    XCII 

Write  down  and  evaluate  the  cofactors  of  each  of  the  elements  of 
the  following  determinants. 


1 

2     3 

1. 

4 

5     6 

7 

8     9 

1 

1      1 

2. 

a 

h      c 

a'^ 

b-' 

j2 

1—1 

1  1 

3. 

n       h       c 

a*     b*     c* 

a     h     g 

4. 

h     b    f 
9    f    c 

208.  The  principal  properties  of  determinants  of  the  third 
order.  It  is  now  easy  to  prove  tlie  following  theorems,  which 
are  quite  analogous  to  the  Corresponding  properties  of  second 
order  determinants.     (Compare  Art.  2U5.) 

1.  A  determinant  of  the  third  order  does  7iot  change  its  value 
if  its  elements  are  transposed,  that  is,  if  its  roivs  he  converted 
into  columns,  and  its  columns  into  rows,  the  relative  order  of 
rotvs  arid  columns  7iot  being  changed. 

To  prove  this,  let 


i)  = 


If  we  expand  D  according  to  the  elements  of  its  first  row, 
and  D'  according  to  the  elements  of  its  first  column,  we  ob- 
tain identical  results,  thus  proving  the  theorem. 

2.  A  third  order  determinant  is  equal  to  zero  if  all  of  the 
elements  of  any  one  of  its  roivs  or  columns  are  equal  to  zero. 

This  follows  directly  if  we  expand  the  determinant  with  respect  to  the 
elements  of  such  a  row  or  column. 

3.  A  determinant  of  the  third  order  changes  its  sign  if  any 
two  of  its  columns  or  any  two  of  its  roivs  are  interchanged.    For 


«1 

h 

^1 

«i 

a.2 

a.. 

h 

^2 

,     D'  = 

h 

h 

a., 

h 

C;, 

Ci 

Co 

instance 


c, 


«i 


^2       Cj 

b,     Co 


as  may  be  shown  by  expanding  both  members  of  this  equation  and  com- 
paring the  two  results. 


Art.  208] 


PROPERTIES   OF   DETERMINANTS 


339 


4.    If  all  of  the  elements  of  a  roiv  (or  column')  are  multiplied 
hy  the  same  number  m,  the  whole  determinant  is  multiplied  by  m. 


Proof.     Let 


D 


ma^ 

iii//^ 

inc^ 

«2 

h 

'"2 

"3 

h 

<^3 

If  we  expand  D  according  to  the  elements  of  the  first  row,  we  have  (see 
Art.  207,  equations  (1))  : 

SO  that 


ma^     mb^     mc^ 

(1.2  f>.,  Co 

"s        h        c. 


—  in  \a.2     It.,     cJ, 
a„     b.,     c. 


thus  proving  the  theorem  for  the  case  where  the  elements  of  the  Jirst  row 
are  multiplied  bj^  a  common  factor.  The  proof  for  any  other  row  or 
column  may  be  carried  out  in  the  same  way. 

5.  A  third  order  determinant  is  equal  to  zero  if  two  of  its 
parallel  lines  (rows  or  columns)  read  alike,  that  is,  if  all  pairs 
of  corresponding  elements  in  two  parallel  lines  are  equal  to  each 
other. 

This  follows  at  once  from  Theorem  2.  For,  let  us  denote  by  D  the  de- 
terminant under  consideration.  If  we  interchange  the  two  lines  which 
read  alike  we  should  find,  according  to  Theorem  2,  a  new  determinant 
D'  such  that 


(1) 


D'  =-  D. 


But  since  the  interchanged  elements  are  equal  in  value,  we  must  also 
have 

(2)  D'  =  D. 

But  from  (1)  and  (2)  we  conclude  by  addition* 

D'  =  D  =  Q). 

6.  If  every  element  of  any  rotv  or  column  is  expressed  as  a 
sum  of  tii'o  terms,  the  determinant  may  be  expressed  as  a  sum 
of  two  determinants,  thns 


Oo  +  m.,     h„     cj  = 
a,  +  ni,     bo     Co  I 


«1 

''. 

'"i 

a.. 

/>., 

c,  + 

«3 

fh 

<^3 

««j  b^  cA 
m.,  b.-,  cJ 
»n„     bo     cJ 


340     LINEAR  EQUATIONS  AND  DETERMINANTS      [Art.  208 


To  prove  this,  expand  the  determinant  on  the  left  according  to  the 
elements  of  the  first  column.  A  similar  proof  will  establish  the  theorem 
for  any  column  or  row. 

7.  If  to  the  elements  of  any  row  or  column  we  add  the  cor- 
respondin(/  elements  of  a  parallel  rotv  or  column^  multiplied  hy 
one  and  the  same  factor,  the  value  of  the  determinant  is  not 
changed. 

For,  by  Theorems  4  and  6  we  may  write 


r/j  +  m/^j. 

^     Cj 

«i 

^^1 

^1 

h 

h 

r/o  +  mh.^, 

h  ^2 

=  a.. 

h 

^2 

+  ??? 

h 

h 

'I3  +  mbg, 

''3  (^3 

";! 

h 

^3 

ih 

h 

and  the  second  determinant  of  the  right  member  is  equal  to  zero  on  ac- 
count of  Theorem  .3.  This  actually  proves  Theorem  7  for  the  first  and 
second  columns.     All  other  cases  may  be  proved  in  the  same  fashion. 

8.  If  the  elements  of  any  rotv  or  column  he  multiplied  hy 
their  respective  cof actors,  the  sum  of  the  products  ohtained  in 
this  ivay  is  equal  to  the  value  of  the  determinant. 

This  is  merely  a  restatement  of  equations  (1)  of  Art.  207. 

9.  If  the  elements  of  any  roiv  or  column  he  multiplied  hy  the 
cofactors  of  the  corresponding  elements  of  a  parallel  row  or 
column,  the  sum  of  these  products  is  equal  to  zero. 

Proof.    We  know,  for  instance,  according  to  Theorem  8,  that 
(3)  a^A^  +  a^A^  +  a^A^  = 


Now  b^A^  +  h^A„  +  ^g.lg  is  obtained  from  tlie  left  member  of  (3)  by  re- 
placing rtj,  flo,  f'3,  by  It^,  l>2,.l>s,  respectively.  Therefore,  /\A^+  l>^A^  +  ^s^^a 
must  be  equal  to  the  determinant 

''2 

which  is  obtained  from  the  right  member  of  (8)  when  we  replace  Oj,  Oj,  Og 
by  61,  62,  63.  Rut  this  determinant  is  equal  to  zero  on  account  of 
Theorem  5.     Therefore 

hiAi  +  biAi  +  hAs-O. 

All  other  cases  of  Theorem  9  may  be  proved  in  the  same  way. 


Art,  208] 


PROPERTIES   OF   DETERMINANTS 


341 


The  theorems  which  we  have  developed  have  many  impor- 
tant applications,  some  of  which  we  shall  explain  in  Art.  209. 
They  may  frec^uently  be  used  to  simplify  the  work  involved 
in  calculating  the  value  of  a  determinant,  as  illustrated  in 
the  following  Exercise. 


(1) 


We  may  write 

4  7     7 

5  -  4    2 
-2  5     1 


EXERCISE    XCIII 

4-2.5,   7-2(-4), 
5  -4 

-2  5 


15     31 

-4     2 

5     1 


siuce  the  second  determinant  in  (1)  may  be  obtained  from  the  first  by 
multiplying  the  elements  of  the  second  row  by  -  2  and  adding  to  the 
corresponding  elements  of  the  first  row  (Theorem  7,  Art.  208).  But 
the  third  determinant  in  (1)  is  equal  to  zero,  since  the  elements  in  its 
first  row  are  exactly  three  times  as  great  as  the  corresponding  elements 
of  its  third  row  (Theorems  4  and  5,  Art.  208).     Therefore 

4,  7,     7 

5,  -  4,     2=0. 
-2,         5,     1 

Show  that  the  following  determinants  are  equal  to  zero : 


1     3       5 

15 

7 

9 

a  +  d     b  +  e     c  +f 

2. 

2     6     10  . 

3.     1 

o 

3 

4. 

a             b             c 

8     9     11 

4 

5 

6 

d            e           f 

5.    Prove  that 

1 

1 

1 

a 

b 

c 

=  (a-b)(b-c)(c-a). 

a"- 

h-^ 

6-2 

Proof.  The  determinant  will  vanish  if  a  =  ft,  or  if  ft  =  c,  or  if  c  =  a 
(Theorem  5,  Art.  208).  Therefore  it  has  a  —  b,  h  —  c,  and  c  —  a  as  factors. 
(See  Factor  Theorem,  Art.  84.)  But  the  product  (a  —  b)(b  —  c)(c  —  n) 
and  the  expanded  determinant  are  integral  rational  functions  of  a,  ft,  r, 
of  the  same  (the  third)  degree.  Therefore  the  determinant  can  differ 
from  this  product  merely  by  a  numerical  factor  L  independent  of  a,  ft, 
and  c. 

Thus,  we  must  have 


(1) 


1      1      1 

n      b      r 
cfi     b'^     c'^ 


L(u  ~b)(b-c)(r  —  a). 


342      LINEAR  EQUATIONS  AND  DETERMINANTS      [Art.  209 


If  the  determinant  be  expanded  it  will  be  found  to  contain  the  term 
hc^.  The  corresponding  term  in  tlie  expanded  right  member  of  (1)  will 
be  Lbc'^.  Therefore  L  must  be  equal  to  1,  which  proves  the  desired 
result. 


6.    Prove  that 


1 

a 

a^ 

1 

h 

h^ 

■1 

1 

c 

,.3 

{a  -  h)(h  -  c)(c  -  a){a  +  h  +  c). 


Prove  that 

oi  +  kao  +  In^ 

02  +  ma^ 

(iz 

a, 

^2 

h  +  ^'bz  +  Ih 

5  2  +  mh^ 

bz 

br 

b.-. 

Cl   +   kC2  +  ICi 

Co  +  mc3 

cz 

^1 

C, 

Find  the  value  of 

a      b 
a2     h^ 

c 

a3     b^ 

c^ 

209.  Solution  of  a  system  of  three  simultaneous  linear 
equations  with  three  unknowns.  We  now  return  to  the 
problem  of  Art.  20G,  to  solve  the  equations 

(1)  (i^x  +  h-^y  +  c^z  =  rp 

(2)  a^x  +  Ky  +  c.^  =  r^, 

(3)  a^x  +  h^y  +  c^z  =  rg, 

for  x^  ?/,  and  z.  It  was  this  problem  which  first  suggested 
to  us  the  introduction  of  determinants  of  the  third  order 
(see  Art.  206),  and  we  shall  now  see  how  easily  the  solution 
of  such  a  system  of  equations  may  be  effected  by  means  of 
determinants. 


Let 


(4) 


D^ 


«1 

^ 

Oo 

b^ 

a.. 

b. 

be  the  determinant  of  the  coefficients  of  x,  y,  and  z,  in  the  equations  (1), 
(2),  and  (.3),  and  let  ^i,  Bu  Ci,  etc.,  be  the  cofactors  of  n\,  by,  c^,  etc.,  in 
the  determinant  D.  (See  Art.  207  for  definition  of  cofactors.)  Let  us 
multiply  both  members  of  (1)  by  Ai,  both  members  of  (2)  by  A2,  both 
members  of  (3)  by  .I3,  and  add.     AVe  shall  find,  after  collecting  terms, 

(ai.li  +  u.Ai  +  a^A3)x+  (biAi  +  M2  +  bzAa)//  +  (r^Ai  +  C2A2  +  ^3^3)2 

=  riAi  +  /•2.42  +  rs^s- 


Art.  209]       THREE   SIMULTANEOUS   EQUATIONS 


343 


The  coeHicieiit  of  ,/•  in  this  equation  is  equal  to  D  (Theorem  8,  Art.  208), 
wliile  the  coeHicieiits  of  y  and  z  are  both  equal  to  zero  (Theorem  9, 
Art.  208).     Tiierefore,  we  find 


(5) 


Dx  =  r\Ai  +  r-zAo  +  nAz. 


If  we  multiply  (1),  (2),  (;J)  by  Bx,  Bo,  B^  respectively  and  add,  we 
find  in  the  same  way 

(6)  nil  =  nBi  +  r.B^  +  r^B^, 
and  by  a  similar  process  we  find 

(7)  Dz  =  riCi+r.>C;  +  rzCz. 

The   right   members  of    (.5),   (6),  and  (7)    may  also   be   written  as 
determinants  of  the  third  order.     In  fact,  if  in  the  equation, 


D 


ax 

hx 

C\ 

a-2 

ho 

C2 

as 

bz 

C3 

axAx  +  a2A2  +  a3A3, 


we  replace  ai,  02,  03,    by  ri,  r-2,  rs,  we  find 

rx     />x     Cx 

r-2     hi     Co   =  rxAx  +  r2A2  + rsAs, 

rs     hs     C3 

which  is  the  right  member  of  (.3).     Similarly  we  conclude  that 


Ol 

ri 

Cl 

('2 

1-2 

<"2 

«3 

^3 

(•3 

riBi  +  7-2^2  +  rsBi, 


«! 

bx 

rx 

ao 

bo 

»'2 

f3 

b3 

n 

rxCx  +  /•2C2  +  raCs. 


(8) 


We  may  therefore  write  equations  (.5),  (0),  and  (7)  as  follows: 

n  bi     Cx 

rj  &2     Co 

rz  b?,     C2 

ai     rx     cx\  ax     bi     Ci 

Co     ro     Co\,  ao     bo     C2 

f'3     ''3     ^'a!  «3     bz     Cz 


rtl 

bi 

''1 

Oo 

b2 

C2 

X  = 

03 

bz 

Cz 

ai 

61 

Cl 

ao 

bo 

C2 

y^ 

«3 

bz 

C3 

ai 

bi 

n 

«2 

bo 

r-2 

as 

bz 

rz 

If  i)  is  different  from  zero,  we  may  divide  both  members 
of  each  of  these  equations  by  D,  and  thus  obtain  the  solution 
of  (1),  (2),  and  (3)  in  the  form  of  three  fractions.  These 
fractions  have  as  their  common  denominator  the  determinant 


344      LINEAR  EQUATIONS  AND  DETERMINANTS      [Art.  210 

D.  The  numerator  of  the  fraction  for  x  is  obtained  from  D 
by  replacing  in  it  the  coefficients  a^,  a^,  ag,  of  x  by  the  right 
members,  r^,  r^,  rg,  of  equations  (1),  (2),  and  (3).  The. 
numerator  of  tlie  fraction  for  y  is  obtained  from  D  by  replac- 
ing in  it  the  coefficients,  hy,  h^^  5g,  of  y  by  r^,  r^,  r^.  The 
numerator  of  the  fraction  for  z  is  obtained  from  D  by  replac- 
ing the  coefficients,  fj,  e^,  r/g,  of  z  by  r^,  7'^,  r^. 


EXERCISE    XCIV 


Solve  the  following  systems  of  equations  by  determinants 


3. 


4. 


5. 


f  2  X  +  ^  +  c  =  -i, 

\x  +  1/  +  2z  =  i. 
i  X  +  ij  +  z  =  SO, 

3  x  +  4  y  +  2  ^  =  50, 
[27  x+  9i/  +  '3z  =  64. 

[  18x  -  Ty  -  57  =  11, 

Hi  y  -  f -^  +  ^  =  1^8, 

\^z  +  2y+  |x=  80. 

7x  —  5z  =  y  +  x  —  86, 
ix  +  ^y  +  lz  =  5S. 
X  +  y  =  \0, 
x  +  z=19, 
y  +  z  =  23. 


6. 


7. 


y  -{-  z  =  a, 
z  +  x  =  h, 
x  +  y  =  c. 

~  +  ~  -  a, 

y    ~ 

Z         X 

1 J  1 

-  H-  -  =  c. 
X      y 


8. 


[■  fl.r  +  hy  =  c, 
,lx  4-  ey  =/, 
^  ////  +  hz  —  I. 
\  ax  +  hy  +  cz  =  d, 
I  n'~.r  +  li-y  +  r~z  =  ^/'^, 
I  r<3.i;  +  h^y  +  c^^  =  d. 

Hint  for  Ex.  9.     Make  use  of  Examples  5  and  8,  Exercise  XCIII. 


210.  Homogeneous  equations.  If  the  equations  (1),  (2), 
(3)  of  Art.  20*J  are  homogeneous,  that  is,  if  r ^  =  rg  =  rg  =  0, 
and  if  their  determinant  D  is  different  from  zero,  equations 
(8)  of  Art.  209  and  Theorem  2  of  Art.  208  show  that  the 
only  solution  of  (1),  (2),  (3)  is  the  obvious  one  x  =  y  =  z  =  0. 
It  would  be  quite  improper  to  draw  the  same  conclusion  in 
the  case  where  D  is  equal  to  zero,  since  equations  (8)  of 
Art.  209  do  not  admit  of  division  by  D  in  this  case.  (See 
Art.  21.)  What  actually  takes  place  in  this  case  is  ex- 
pressed by  the  latter  part  of  the  following  theorem,  which 
covers  both  the  case  when  i)  ^  0,  and  i>  =  0. 


Akt.  210]  HOMOGENEOUS   EQUATIONS  345 

The  Jiomogeneous  equations 

(1)  a^j-  +  />!//  +  e^z  -  0, 

(2)  a^j'  +  ^.3//  +  ^2.'  =  0, 

(3)  a^.r  +  b^>/  +  CgS  =  0, 

Aaug  7<o  solution  except  x  =  //  =  z  =  0,  /f  f 7i«!i/*  determinant  is 
different  from  zero.  But  if  the  determinant  is  equal  to  zero., 
they  have  infinitely  mant/  solutions. 

Proof,  ^^'e  have  settled  the  case  D  =fz  O  already.  Let  us  as.siime 
therefore  that  D  is  equal  to  zero,  but  that  not  all  of  the  second  order 
minors  of  D  are  equal  to  zero.  There  will  then  exist  at  least  one  second 
order  minor  of  D  which  is  not  equal  to  zero.  Let  us  assume  in  partic- 
ular that 


t>2       ^2 

b.,     C3 


does  not  vanish.  We  may  then  solve  equations  (2)  and  (3)  for  y  and 
z  in  terms  of  r,  as  in  Art.  206,  thus  obtaining  equations  (5)  of  Art.  20G, 
where,  however,  we  now  have  to  put  ?-j  =  r.,  =  0.  If  we  take  account  of 
this,  and  make  use  of  the  notation  introduced  in  Art.  207  for  the  co- 
factors  of  the  various  elements  of  D,  we  find  that  these  equations  (the 
equations  ("))  of  Art.  206)  reduce  to 

A^y  =  B^r  and  A  f  =  C\x, 
whence 

(4)  2/  =  ^-^-^      ^  =  ^^- 

.4,  A, 

These  values  of  1/  and  z  will  satisfy  equations  (2)  and  (3),  no  matter 
what  value  be  given  to  x.  But  they  will  also  satisfy  (1).  For  if  we 
substitute  the  values  (4)  of  //  and  ~  into  the  left  nieni])er  of  (1),  we  find 

a^x  +  h^ii  +  c^z  —  a^x  +  h^^x  +  r,  ^  '  j; 
- '  1  •  M 

=  —  {a\A  1  -1-  bilii  +  <\C\)x  -  — .r, 
-'1  '1 

and  this  is  equal  to  zero,  no  matter  what  value  x  may  have,  since  D  is 
equal  to  zero  by  hypothesis. 

Since  equations  (4)  give  a  solution  of  (1),  (2),  (3),  no  matter  what 
value  be  given  to  x,  we  see  that  there  exist  in  this  case  infinitely  many 
solutions  of  the  tlii-ee  given  equations,  as  our  theorem  asserts. 

Our  proof  was  based  011  the  assumption  that  A^  is  not 
equal  to  zero.     It  is  easy  to  see  how  the  proof  should  be 


346     LINEAR  EQUATIONS  AND  DETERMINANTS      [Art.  211 

modified  if  A^  is  equal  to  zero,  provided  that  there  exists 
some  otlier  second  order  minor  of  D  which  is  not  equal  to 
zero.  We  may,  in  fact,  state  our  theorem  a  little  more 
precisely  as  follows: 

If  the  determinant  D  of  the  homogeneous  equations  (1),  (2), 
(3)  is  equal  to  zero,  hut  if  there  exists  at  least  one  second  order 
minor  in  D  which  does  not  vanish,  theji  there  exist  infinitely 
many  solutions  of  (1),  (2),  (3),  hut  the  ratios  of  the  three  un- 
knowns, X,  y,  z,  will  he  determined  uniquely  hy  these  equations. 

For  instance,  if  ^4^  ^  0,  equations  (4)  show  that  these  ratios  are 
y  -.x  -  B^.A^,   z:  X  -  C-^-.yi-^, 
x:y:z  =  A^:B,:C\. 

If  i)  =  0,  and  if,  besides,  all  of  the  second  order. minors  of 
D  are  equal  to  zero,  it  may  be  shown  that  two  of  the  three 
unknowns,  x,  y,  z,  may  be  chosen  arbitrarily.  In  this  case 
the  ratios  x  :  y  :  z  are  not  determined  uniquely. 

Our  method  of  proof  showed  us  that,  in  the  case  D  =  0, 
any  solution  of  two  of  the  three  equations  would  also  satisfy 
the  third.  We  may  express  this  by  saying  that  the  three 
equations  are  not  independerit  ivhen  D  is  equal  to  zero.  In  the 
case  when  J)  =  0,  and  when,  besides,  all  of  the  second  order 
minors  of  D  are  equal  to  zero,  any  solution  of  one  of  the 
equations  satisfies  both  of  the  others.  In  this  case,  then,  two 
of  the  three  equations  (1),  (2),  (3)  are  mere  multiples  of  the 
third,  and  the  three  equations  taken  together  convey  no  more 
information  about  the  unknown  quantities  than  a  single 
equation. 

211.  An  application  of  linear  equations  in  chemistry.  There 
are  certain  substances  like  iron,  silver,  lead,  sulphur,  hydro.- 
gen,  oxygen,  chlorine,  etc.,  which  chemists  have  not  been, 
able  to  separate  into  other  constituents,  and  which  they  call 
elements.  Very  frequently  several  elements  occur  in  a  mix- 
ture which  may  contain  its  various  constituent  parts  in  any 
proportion.     Thus  pulverized  iron  and  pulverized  sulphur 


Art.  211]       LINEAR   EQUATIONS   IN   CHEMISTRY  347 

may  be  mixed  in  aii}'  [H'oportion,  but  the  various  particles 
are  still  recognizable  as  iron  or  sulphur  particles.  If  heat  be 
applied,  some  of  the  iron  and  sulphur  particles  will  combine 
to  form  a  new  substance  which  is  essentiall}^  different  in  kind 
from  both  of  its  component  elements,  and  which  is  said  to  be 
a  chemical  combination  or  compound  of  iron  and  sulphur. 

It  is  a  fundamental  fact  of  chemistry  that  the  elements  enter 
into  chemical  combinations  only  in  certain  fixed  proportions, 
while  in  a  mixture  the  proportions  may  be  any  whatever. 

For  instance,  one  gram  of  hydrogen  (chemical  symbol  H) 
will  combine  with  35.4  grams  of  chlorine  (chemical  symbol  CI) 
to  form  86.4  grams  of  hydrochloric  acid  (HCl).  If  the  vessel 
in  which  the  reaction  takes  place  should  contain  just  one 
gram  of  hydrogen  and  more  than  the  proper  amount  of 
chlorine,  only  35.4  grams  of  the  chlorine  will  be  used  up. 
The  rest  will  remain  unchanged.  This  reaction  is  repre- 
sented by  the  chemical  equation 

H  +  CI  =  HCl. 

To  account  for  the  law  of  fixed  proportions,  the  English 
physicist  and  chemist  Dalton  (1766-1844)  proposed  the 
following  theory.  A  gram  of  hydrogen  contains  a  certain 
very  large  number  of  exceedingly  minute  particles  called 
atoms.  Similarly,  the  chlorine  in  the  vessel  is  composed  of 
atoms  of  chlorine.  Each  atom  of  hj-drogen  combines  with 
one  atom  of  chlorine  to  form  a  smallest  particle,  called  a 
molecule,  of  the  compound.  If  all  of  the  material  is  used  up, 
there  is  one  atom  of  hydrogen  combined  with  every  atom  of 
chlorine,  and  the  law  of  fixed  proportion  will  be  accounted 
for  if  we  assume  that  all  hydrogen  atoms  have  the  same 
weight,  and  that  every  chlorine  atom  weighs  35.4  as  much  as 
a  hydrogen  atom. 

Every  chemical  reaction  is  capable  of  a  similar  interpreta- 
tion, and  as  a  result  of  a  systematic  study  of  all  the  elements 
and  their  compounds,  chemists  have  come  to  attribute  to 
every  element  a  number,  called  its  atomic  weight.  The 
atomic  weight  of  an  element  tells*  us  how  the  weight  of  an 


348      LINEAR  EQUATIONS  AND  DETERMINANTS      [Ain.  211 

atom  of  that  element  compares  with  that  of  a  hydrogen  atom. 
Thus  the  atomic  weight  of  chlorine  is  35.4,  as  determined  by 
the  reaction  which  was  described  above. 

These  atomic  weights  are  sometimes  called  relative  atomic 
weights.  They  do  not  teach  us  how  heavy  any  atom  really 
is.  When  we  say  that  an  element  has  the  relative  atomic 
weight  jw,  Ave  only  mean  that  one  of  its  atoms  weighs  p  times 
as  much  as  a  hydrogen  atom.  If  the  absolute  weight  of  a 
hydrogen  atom  (expressed  in  grams)  is  w  grams,  the  absolute 
weight  of  an  atom  of  this  other  kind  will  then  be  pw  grams. 

Let  us  now  consider  three  elements  A^  B,  and  P,  whose 
relative  atomic  weights  are  a,  b,  and  p  respectively,  and  let 
AP  and  BP  be  two  chemical  combinations  the  molecule  of 
each  of  which  contains  only  one  atom  of  each  constituent. 
Suppose  that  we  have  a  mixture  of  the  two  compounds  AP 
and  BP,  and  that  we  know  the  total  weight  g  (in  grams) 
of  the  mixture,  and  also  the  weight  h  of  the  element  P  which 
is  present  in  the  mixture.  We  can  then  calculate  the  amount 
present  in  the  mixture  of  each  of  the  two  compounds  AP  and 
BP. 

The  solution  of  this  problem  is  known  as  indirect  analysis 
of  the  mixture,  and  is  often  of  practical  importance.  For  it 
is  easy  to  hud  g  the  total  weight  of  the  mixture,  and  in  many- 
cases  it  is  easy  to  lind  the  amount  h  of  the  element  P,  while 
it  may  be  very  difficult  to  separate  the  elements  A  and  B  by 
chemical  means  and  to  determine  their  weights  directly. 

To  solve  the  problem  we  proceed  as  follows :  Let  us 
denote  by  x  and  y  the  number  of  grams  present  in  the  mix- 
ture of  the  compounds  AP  and  BP  respectively.  Then  of 
course 

(1)  ^  +  y  =  9^ 

since  g  represents  the  total  weight  of  the  mixture. 

Let  w  be  the  weight  (in  grams)  of  a  hydrogen  atom. 
Then  an  atom  of  A  weighs  aw  grams,  one  of  B  weighs  hw 
grams,  and  one  of  P  weighs  pw  grams.  Consequently  a 
molecule  of  AP  weighs  at<;+jow  or  (a  +  jt?)M'grams,  and  there 


Art.  211]       LINEAR   EQUATIONS   IN   CHEMISTRY  349 

are  x  I  {a  +  p^iv  molecules  of  AP  and  7/  /  {b  +  p)w  molecules 
of  BP  in  the  mixture.  Again,  since  there  are  h  grams  of 
the  element  P  in  the  mixture  and  since  each  atom  of  P 
weighs  pw  grams,  there  are  h/pw  atoms  of  P  in  the  mixture. 
But  there  is  one  and  only  one  of  these  atoms  in  every  molecule 
of  AP  and  BP.  Therefore  the  number  of  molecules  of  ^P 
and  BP  taken  together  must  be  equal  to  the  number  of  atoms 
of  P ;   that  is 

X      _,_      y      ^  A 

{a-\-p^w      {b  +  p)w     pw^ 
whence,  multiplying  both  members  by  w, 

(2)  ^    1-   y    =^. 

a+p      b+p     p 

Thus,  we  have  found  two  equations,  (1)  and  (2),  for  our  two 
unknowns.     We  may  write  them  as  follows, 

(3)  I  ^  +  y  =  9^ 

\p(b  +p)x-\-p(a  +p)i/=(a+p^(b  +p}h. 
The  determinant  of  the  coefficients  of  x  and  1/  is 
1  1 

p{h+p').  lK(^+p) 
Consequently  this  determinant  will  be  different  from  zero 
whenever  a  is  different  from  J,  and  it  will  actually  be  possi- 
ble to  solve  (3)  for  the  two  unknowns.     The  result  is 

p  a—  b  p  a  —  b 

Illustrative  example.  A  mixture  composed  of  sodium  chloride  (NaCl) 
and  potassium  chloride  (KCl)  was  found  to  weigh  3  grams.  The  com- 
pounds were  then  decomposed  and  tlie  amount  of  chlorine  was  found  to 
be  1.7  grams.  Find  the  amount  of  NaCl  and  of  KCl  present  in  the 
original  mixture. 

The  atomic  weights  in  this  case  are  a  =39.1  for  potassium  (K),  i  =  23 
for  sodium  (Na),  and  p  =  35.4  for  chlorine  (CI).  We  liave  also  g  —  S, 
h  =  1.7.  Substitution  of  these  numbers  in  formulas  (i)  gives  (to  the 
nearest  tenth  of  a  gram) 

X  =  0.9  gram,    //  =  2.1  grams, 
so  that  there  were  0.9  gram  of  KCl  and  2.1  grams  of  NaCl  in  the  mixture. 


--p(ia+p)  —  p{b+p)  =  p(a-  5). 


350      LINEAR  EQUATIONS  AND  DETERMINANTS      [Art.  212 

212.  Generalization  to  systems  of  n  linear  equations  with  n 
unknowns.  The  attempt  to  generalize  the  methods  of  this 
chapter  to  the  case  of  n  equations  with  n  unknowns,  leads 
to  the  introduction  of  determinants  of  the  nth  order.  This 
might  be  done  by  the  method  of  mathematical  induction. 
(See  Arts.  84  and  88.)  But  a  more  elegant  treatment  of 
this  subject  may  be  based  on  the  theory  of  permutations, 
which  is  also  important  from  many  other  points  of  view. 
We  shall  therefore  reserve  the  discussion  of  w  linear  equa- 
tions with  n  unknowns  for  Chapter  XII,  devoting  the  next 
two  chapters  to  some  of  the  numerous  questions  concerned 
with  permutations  and  combinations  and  the  theory  of 
probability. 

EXERCISE    XCV 

1.  Find  two  numbers  whose  sum  is  equal  to  60  and  -whose  difference 
is  equal  to  16. 

2.  If  the  first  of  two  numbers  is  multiplied  by  2,  the  second  by  .5, 
the  sum  of  the  products  is  31 ;  if  the  first  be  multiplied  by  7,  the 
second  by  4,  the  sum  of  the  products  is  equal  to  68.  Find  the  two 
numbers. 

3.  A  father  was  three  and  one-third  times  as  old  as  his  son  six 
years  ago.  Three  years  from  now  his  age  will  be  two  and  one-sixth 
times  the  age  of  his  sou.     What  are  the  present  ages  of  father  and  son? 

4.  A  father  was  m  times  as  old  as  his  son  six  years  ago.  Three 
years  from  now  his  age  will  be  n  times  the  age  of  his  son.  "What  are 
the  present  ages  of  father  and  son  ?  Shall  we  obtain  a  reasonable  answer 
no  matter  what  the  values  of  m  and  n  may  be  ? 

5.  A  tank  containing  21,000  liters  may  be  filled  by  means  of  two 
pipes.  If  the  first  pipe  is  opened  for  4  hours  and  the  second  for  5  hours, 
9000  liters  of  water  are  obtained.  If  the  first  pipe  is  o]ien  for  7  hours  and 
the  second  for  ^  hours,  12,600  liters  are  obtained.  What  is  the  flow,  in 
liters  per  hour,  of  each  pipe,  and  how  long  will  it  take  to  fill  the  tank 
if  both  pipes  are  opened  at  the  same  time  ? 

6.  It  is  found  that  21  kilograms  of  silver  weigh  2  kilograms  less 
in  water  than  in  air,  and  that  0  kilograms  of  copper  lose  1  kilogram 
when  weighed  in  water.  An  alloy  of  silver  and  copper  weighing  118  kilo- 
grams in  air  is  found  to  weigh  14|  kilograms  less  in  water.  How  much 
silver  and  copper  are  there  in  the  alloy?     (See  Art.  112.) 


Art.  212]       GENERALIZATION    TO   :M0RE    UNKNOWNS       351 

7.  Generalization  of  Ex.  6.  p  kilograms  of  metal  yl  lose  n  kilograms 
in  water,  p  kilograms  of  metal  B  lose  b  kilograms  in  water.  An  alloy 
of  the  two  metals,  A  and  B,  weighing  p'  kilograms  in  air,  loses  c  kilo- 
grams in  water.     How  much  of  each  element  is  there  in  the  alloy? 

8.  According  to  the  story  told  by  Vitruvius  (see  Art.  112),  the  crown 
of  King  Hiero,  composed  of  gold  and  silver,  weighing  twenty  pounds  in 
air,  lost  1.25  pounds  when  weiglied  in  water.  liut  1!).(J1  jJOiinds  of  gold 
and  10.')  pounds  of  silver  each  lose  one  pound  in  water,  AVhat  was  the 
composition  of  the  crown? 

9.  Two  trackmen  are  practising  on  a  circular  track  126  yards  in 
circumference.  When  running  in  opposite  directions,  they  meet  every 
13  seconds.  Running  in  tiie  same  direction,  the  faster  passes  the  slower 
every  126  seconds.  How  many  minutes  does  it  take  each  of  the  men  to 
run  a  mile  ? 

10.  The  planet  Mercury  makes  a  circuit  about  the  sun  in  3  months. 
Venus  makes  a  circuit  in  1\  months.  Find  the  number  of  months  be- 
tween two  successive  times  when  Mercury  is  between  Venus  and  the  sun. 

11.  Find  three  numbers  whose  sums,  taking  two  of  them  at  a  time, 
are  equal  to  a,  h,  c  respectively. 

12.  Three  brick  masons.  A,  B,  and  C,  are  building  a  wall.  A  and  B 
alone  would  require  12  days  to  complete  the  job ;  B  and  C  would  require 
20  days;  and  A  and  C  would  require  15  days.  How  much  time  would 
be  required  for  each  working  alone,  and  how  long  will  it  take  them  to 
build  the  wall  if  all  of  them  work  together? 

13.  Generalize  Example  12  by  substituting  a  for  12,  h  for  15,  c  for  20. 

14.  A  cistern  is  filled  by  three  pipes.  A,  B,  and  C.  A  and  B  together 
will  fill  it  in  70  minutes,  A  and  C  in  84  minutes,  B  and  C  in  140  minutes. 
How  long  will  it  take  each  pipe  separately  to  fill  the  cistern  ?  How  long 
when  all  of  them  are  running  at  once? 

15.  A  first  mass  of  alloy  contains  5  oz.  of  gold,  15  oz.  of  silver,  and  30  oz. 
of  copper.  A  second  mass  contains  20  oz.  gold.  28  oz.  silver,  48  oz.  copper. 
A  third  alloy  is  composed  of  12  oz.  gold,  39  oz.  silver,  24  oz.  copper. 
How  much  must  we  take  of  each  of  these  alloys  in  order  to  obtain  a 
fourth  alloy  composed  of  10  oz.  gold,  25  oz.  silver,  and  26  oz.  copper? 

16.  A  certain  number  contains  three  digits  in  arithmetical  progres- 
sion. If  the  number  he  divided  by  the  sum  of  its  digits,  the  quotient  is 
48.  If  108  be  subtracted  from  the  number,  the  remainder  will  have  the 
same  digits  as  the  original  number,  but  arranged  in  the  opposite  order. 
Find  the  number. 


CHAPTER   X 

PERMUTATIONS   AND   COMBINATIONS 

213.  The  notion  of  order.  If  we  have  n  elements  of  any 
kind  such  as  numbers,  letters,  chairs,  tables,  animals,  or  per- 
sons, we  may  think  of  these  n  elements  as  being  arranged 
along  a  straight  line.  In  any  such  arrangement  we  shall 
speak  of  one  of  the  elements  as  the  first,  another  as  the 
second,  and  so  on.  Any  arrangement  of  this  sort  differs 
from  any  other  arrangement  of  the  same  elements  merely  in 
the  order  in  which  the  various  elements  are  thought  of,  not 
in  the  total  number  of  elements  included  in  the  arrangement. 

214.  Permutations.  Each  of  the  various  ordered  arrange- 
ments tvhich  can  be  made  of  n  elements  is  called  a  permutation 
of  these  elements.  The  principal  problem  which  we  shall 
have  to  solve  is  this  :  how  many  permutations  are  there 
of  n  elements,  if  each  of  the  elements  occurs  in  each 
permutation  ? 

Clearly  a  single  element  can    be  arranged  in  one  way  only.     If  we 
have  two  elements,  let  us  represent  them  by  the  letters  a  and  h.*     There 
are  clearly  two  arrangeinents  and  only  two  in  this  case;  namely  a  first 
one  with  a  in  the  first  place  and  h  in  the  second,  and  a  second  arrange- 
ment with  h  in  the  first  place  and  a  in  the  second.     We  may  represent 
these  two  arrangements  symbolically  as  follows  :  ab  and  ha. 
In  the  case  of  three  elements,  a,  b,  and  c,  there  are 
Two  arrangements,  ahc  and  ach,  in  which  a  occupies  the  first  place. 
Two  arrangements,  bac  and  bco,  in  which  b  occupies  the  first  place. 
Two  arrangements,  cab  and  dm,  in  which  c  occupies  the  first  place, 
or  3  . 2  =  6  arrangements  altogether.     Thus  there  are  six  permutations 
of  three  elements. 

*  This  is  somewhat  of  a  departure  from  our  general  practice.  So  far  we  have 
used  the  letters  a  and  b  only  to  stand  for  numbers.  In  this  connection,  however, 
a  and  b  may  represent  two  different  persons,  two  animals,  or  two  elements  of  any 
kind. 

352 


Art.  214]  PERMUTATIONS   OF   n   ELEMENTS  353 

Let  us  consider  the  case  of  four  elements,  a,  b,  c,  d.  By  our  previous 
argument  there  are  six  permutations  in  which  a  occupies  the  first  place. 
For,  after  a  has  been  put  into  tlie  first  place  there  remain  three  other 
elements,  6,  c,  r/,  which  may  be  permuted  among  the  remaining  places. 
Similarly  there  are  six  permutations  witli  h  in  the  first  place,  six  with  c 
in  the  first  place,  and  six  with  tl  in  the  first  place,  or  4  x  6  =  24  permu- 
tations in  all. 

Let  us  now  pass  to  the  general  case.  We  denote  the  n 
elements  by 

(1)  a^,  a^.  «3,  .••  a„, 

and  we  use  the  symbol  P„  to  denote  the  number  of  permutations 
of  these  n  elements. 

If  we  omit  any  one  of  the  elements  (1)  there  are  n  —  1 
left,  and  the  number  of  permutations  of  these  n—  1  elements 
will  be  represented  by  the  symbol  P„_j.  Now  all  of  the 
permutations  of  a^,  a^.  •••,  a^  may  be  divided  into  n  classses 
according  as  a^,  or  «2i  ^^i'  «3'  ••••>  or  a„,  occupies  the  first  place. 
The  number  of  permutations  in  any  one  of  these  classes  is 
represented  by  tlie  symbol  Pn-\-  I' or,  the  first  element 
having  been  fixed,  there  are  onl}^  n  —  1  elements  left  to  be 
arranged.  Since  there  are  n  such  classes,  there  will  be  n 
times  Pn-i  permutations  in  all.  But  we  have  denoted  the 
total  number  of  permutations  of  n  elements  by  P„.  There- 
fore we  must  have 

(2)  P,,  =  nP^_y 

We  have  found  already 
Pj  =  l,   P2  =  l-2,   ^3=  1.2.  3  =  6,   P4  =  1.2.3.4  =  24. 

By  using  (2)  we  now  find 

P^  =  .5P4  =  1  •  2  .  3  .  4  .  5  =  120,  Pg  =  1.2. 3. 4. 5. 6  =  720, 

suggesting  the  general  formula 

(3)  P„  =  l  .2.3.4. ..w. 

The  product  of  all  of  the  integers  from  1  to  w,  which  makes 
its  appearance  here,  is  called  factorial  n,  and  is  usually  de- 


354  PERMUTATIONS   AND   COMBINATIONS     [Art.  215 

noted  by  one  of  the  two  symbols  n !  or  \n.     Thus  (3)  be- 
comes 
(4)       Pn  =  n  !  (read  P  sub  n  is  equal  to  factorial  ti). 

To  complete  the  proof  of  this  formula,  we  use  the  method 

of  mathematical  induction.     We  know  that  (4)  is  correct 

for  w  =  1,  2,  3,  4,  5,  6.     We  can  prove  that  if  (4)  is  correct 

for  n  =  k^  then  it  ivill  also  be  correct  for  n=  k+  1. 

In  fact,  if 

P,  =  ;^!  =  1.2.3...yt, 

then,  according  to  (2), 

P,^^=(k+l)P,    =    kl(k+l)=1.2.S:.k.k+l={k    +    l^U 

tlius  proving  our  assertion,  and  consequently  the  validity  of 
(4)  for  all  values  of  w. 

215.  The  number  of  permutations  of  n  elements  taken  k  at 
a  time.  Suppose  again  that  we  have  w  elements  a^,  ag^  •*•  ^w 
and  let  us  examine  in  how  many  ways  these  may  be  arranged 
in  groups  of  k  elements  each,  where  k  S  w?  attention  being 
given  to  the  order  of  the  arrangement  in  each  of  the  groups 
of  k  elements. 

The  following  is  a  concrete  problem  of  this  sort.  There  are  25  base- 
ball players  in  a  college  (n  =  25).  Each  of  them  is  willing  to  take  any 
position  on  the  college  team  composed  of  nine  players  {k  =  9).  In  how 
many  ways  can  the  team  be  formed? 

Each  of  the  required  permutations  is  composed  of  k  ele- 
ments. The  first  place  may  be  filled  in  n  ways,  since  it 
may  be  occupied  by  any  one  of  the  n  elements.  After 
the  first  place  has  been  filled  there  are  only  n  —  1  ways  of 
filling  the  second  place.  Thus  the  first  two  places  may 
be  filled  in  w(w  —  1)  different  ways.  There  are  7i  —  2 
elements  still  available  for  filling  the  third  place,  so  that 
the  first  three  places  may  be  filled  in  n{n  —  1)  (w  —  2) 
ways.  If  we  continue  this  argument  we  see  that  the  k 
places  may  be  filled  in  n(n  —  l)(w  —  2)  •••  (ri  —  A;  +  1)  dif- 
ferent ways. 


Art.  215]  n   ELEMENTS  k  AT   A   TIME  355 

Let  us  now  use  the  symbol  „Pt  to,denote  the  number  of  per- 
mutations of  n  elements  taken  k  at  a  time.      We  have  found 

(1)  „P,  =  n(w  -  l)(w  _  2)  ...  (n  -  y^  +  1), 

the  right  member  being  a  product  of  k  factors,  namely  of 
n  —  0,  n  —  1,  n  —  2,  •••  n  —  (k  —  1). 

Of  course  for  k  =  n,  we  find,  as  in  Art.  214, 

Since  we  may  write 
nl=n(n-l}(H-2^  ...  (n-k  +  l)(n-^^(n-k-l)  ...2.1, 

and  since  the  product  of  the  first  k  factors  in  the  right 
member  is  „Pfc,  while  the  product  of  the  remaining  n  —  k 
factors  is  Qn  —  ^) !,  we  have 

n\  =  ^P,(n-ky., 
whence 

n ! 


(2)  .  „P.  = 


(n-k)\ 


EXERCISE     XCVI 

1.  In  how  many  ways  can  eight  soldiers  be  arranged  in  a  row  ? 

2.  How  many  permutations  of  the  letters  a  h  c  d  e  f  g  are  there 
if  each  of  the  letters  occurs  in  each  permutation  ? 

3.  Of  the  permutations  mentioned  in  Ex.  2,  how  many  begin  with 
a?     How  many  begin  with  aft?  with  abc'i  with  ahcd'i 

4.  How  many  of  the  permutations  mentioned  in  Ex.  2  contain  the 
letters  ahcd  consecutively  and  in  this  specific  order? 

5.  How  many  different  permutations  are  there  of  the  letters  of  the 
word  "  stone  "  when  tliree  are  taken  at  a  time? 

6.  There  are  15  baseball  players  in  a  college,  each  of  whom  is  will- 
ing to  take  any  position  on  the  college  team.  In  how  many  ways  can 
a  team  be  formed  ? 

7.  How  many  of  the  niimbers  between  10  and  100  contain  two  dis- 
tinct digits,  not  counting  zero  as  a  digit? 

8.  How  many  numbers  of  tin-ee  different  digits  can  be  formed  from 
the  seven  digits  1,  2,  •••,  7? 


356  PERMUTATIONS   AND   COMBINATIONS     [Art.  216 

9.    With  eight   flags  of   different  color,  how  many  signals  can  be 
formed  by  displaying  four  of  them  at  a  time? 

10.  How  many  permutations  are  there  of  n  things  taken  r  or  fewer 
than  r  at  a  time,  that  is,  if  it  be  admitted  that  we  may  select  only  one, 
or  two,  or  three,  •••,  or  as  many  as  r  of  these  things? 

11.  How  many  signals,  composed  of  one,  two,  or  three  flags  can  be 
formed  from  five  different  flags  ? 

216.  Circular  arrangements.  The  notion  of  order  which 
we  introduced  in  Art.  213  may  be  called  more  specifically 
linear  order,  since  it  is  suggested  by  the  arrangement  of  n 
objects  placed  in  a  row,  or  on  a  straight  line.  If  the  n 
objects  are  instead  placed  on  the  circumference  of  a  circle, 
or  on  any  other  simple  closed  curve,  we  may  adopt  the  point 
of  view  that,  in  such  a  circular  arrangement,  it  is  unnatural 
to  fix  upon  any  one  of  the  elements,  rather  than  upon  any 
other  one,  as  being  the  first. 

Thus,  in  Fig.  69,  we  may  consider  that  the  three  points 
A,  B,  C  have  the  same  circular  order  whether  we  start  from 
A  and  then  proceed  in  the  order  ABC,  or 
wliether  we  start  from  B  and  proceed  in 
order  to  0  and  A,  or  finally  whether  we 
start  from  O  and  proceed  in  the  order  CAB. 
Thus,  although  there  are  six  permutations  of 
these  three  elements,  there  are  only  two  cir- 
cular permutations.  In  one  of  these  permu- 
tations we  go  around  the  circle  in  clockwise  fasliion  ;  and 
the  other  order  may  be  described  as  a  counter-clockwise 
arrangement. 

In  general,  if  we  wish  to  arrange  n  elements  in  circular  order, 
we  may  place  any  one  of  the  elements  in  a  fixed  position  and 
leave  it  there.  The  remaining  elements  can  then  be  arranged 
in  (rt— 1)!  different  ways.  Therefore,  there  are  (w  —  1)  ! 
circular  permutations  of  n  elements.  In  some  cases  there  are 
pairs  of  clockwise  and  counter-clockwise  arrangements  which 
are  indistinguishable;  namely,  whenever  it  is  admissible  to 
turn  the  circle  around  a  diameter  through  an  angle  of  180°. 
In  such  cases  the  number  of  arrangements  reduces  to  ^(w—  1)  I. 


Art.  217]  ELEMENTS   NOT   ALL   DISTINCT  357 

EXERCISE    XCVll 

1.  lu  placing  a  party  of  people  at  a  round  table,  two  arrangements 
are  regarded  as  equivalent  which  give  each  person  the  same  left  and 
right  hand  neighbors.  How  many  different  ways  are  there  of  seating  8 
people  at  a  round  table  ? 

2.  Seven  beads  of  different  colors  are  to  be  strung  on  a  closed  wire. 
In  how  many  ways  may  this  be  done  ? 

3.  Six  distinct  points  have  been  selected  on  a  closed  curve  (an  ellipse 
for  instance).  How  many  different  hexagons  can  be  formed  with  these 
six  points  as  vertices?  (Hexagons  whose  perimeters  intersect  themselves 
are  admissible.) 

4.  In  how  many  orders  can  a  liost  ami  seven  guests  sit  at  a  round 
table  so  that  the  host  may  have  the  guest  of  highest  rank  upon  his  riglit, 
and  the  next  in  rank  on  his  left? 

5.  If  we  have  «  beads  of  different  colors  to  form  a  bracelet,  how  many 
distinguishable  arrangements  are  possible? 

217.    Permutations  when  all  of  the  elements  are  not  distinct. 

It  happens  quite  frequently  that  the  elements  which  are  to 
be  arranged  are  not  distinct.  Such  is  the  case,  for  instance, 
if  we  wish  to  answer  the  following  question.  How  many 
distinct  words  of  eight  letters  each  can  be  formed  from  the 
letters  i,  Z,  ?,  i,  n,  o,  ^,  s  ?  There  will  not  be  as  many  as  8  !, 
because  several  of  the  8  I  permutations  of  these  eight  letters 
will  give  rise  to  the  same  word,  since  there  are  three  I's  and 
two  Vs. 

The  general  question  is  easily  answered.  Among  the  n 
symbols  which  are  to  be  arranged,  let  us  suppose  that  a 
occurs  r  times,  b  occurs  s  times,  c  occurs  t  times,  and  so  on. 
We  find  it  convenient,  for  the  purpose  of  our  proof,  to  write 
aj,  rTg,  ^3,  •••,  a^  for  the  r  symbols  a ;  b-^,  b^.  •••,  b^  for  the  s  sym- 
bols b  ;   and  so  on.     Thus  we  have 

r  symbols  a^,  a^,  •••,  a^,  each  of  which  means  the  same  as  a, 
(1)8  symbols  b^,  b^,  •••,  6,,  each  of  which  means  the  same  as  b, 
t  symbols  c^,  c^^  •••,  c^,  each  of  which  means  the  same  as  c, 
etc.  etc. 

Since  there  are  w  symbols  all  together  (counting  repetitions), 
we  have  n  =  r  +  s +  t  -\-  ■■■.^ 


358  PERMUTATIONS   AND   COMBINATIONS      [Art.  217 

and  there  are  n !  permutations  of  w  symbols.  But  not  all 
of  the  n !  permutations  will  be  distinct.  Let  X  be  the 
number  of  these  permutations  which  are  distinct,  and  let 
us  imagine  that  all  of  these  X  permutations  have  been 
written  down. 

From  each  of  these  X  distinct  permutations  we  can  obtain 
r !  permutations  by  interchanging  the  r  symbols  a,  leaving 
the  5's,  c's,  etc.,  fixed.  We  obtain  in  this  way  X  •  r\  arrange- 
ments. From  each  of  these  we  can  obtain  s  !  arrangements 
by  permuting  the  6's,  leaving  the  «'s,  c's,  etc.,  fixed.  This 
gives  rise  to  X ■  r\  s\  arrangements.  Proceeding  in  this 
way,  we  find  that  the  total  number  of  arrangements  will  be 

Xrlsltl-.'      . 

But,  since  there  are  n  symbols  all  together,  the  total  number 
of  arrangements  is  also  equal  to  m  !.     Therefore 

X 'r\s\t\  -"  =n\,  or 


(2)  X  = 


r\s\t\  ..• 


Consequently,  (2)  gives  the  number  of  distinct  permutations 
of  n  elements,  r  of  which  are  a's,  s  of  which  are  b's,  t  of  which 
are  c''s,  and  so  on. 

EXERCISE    XCVIII 

1.  How  many  permutations  can  be  made  of  the  letters  of  the  word 
Illinois  ? 

2.  How  many  permvitations  can  be  made  of  the  letters  of  the  word 
Mississippi  V 

3.  A  desk  has  r  pigeonholes;  n  documents  are  to  be  filed  in  these 
pigeonholes  so  that  a  of  them  sliall  go  into  the  first,  jS  into  the  second, 
y  into  the  third,  and  so  on.     In  how  many  ways  may  this  be  done? 

Remark.  The  pigeonholes  are  distinguished  from  each  other  as  first, 
second,  and  so  on.  But  it  is  regarded  as  indifferent  in  what  order  the 
documents  are  arranged  inside  of  the  holes. 

4.  How  many  different  numbers  of  seven  digits  each  can  be  formed 
by  permuting  the  figures  1112225? 


Aim.  218]         TWO   CLASSES   OF   PERMUTATIONS  359 

5.  Piove  the  following  theorem.  //  there  are  n  distinct  elements 
Ou  «2>  •••)  o„,  which  are  to  be  arranged  in  sets  of  r  elements  at  a  time,  and  if 
it  be  permitted  that  each  element  be  repeated  as  often  as  r  times,  then  there 
are  n'' permutations. 

218.  Two  classes  of  permutations.  It  happens  frequently 
that  some  particular  arrangement  of  7i  elements  is  regarded 
as  more  important  or  more  natural  than  any  other.  Thus, 
if  the  elements  under  consideration  are  the  n  numbers  1,  2, 
3,  •••,  w,  we  naturally  think  of  the  smallest  number  first  and 
then  arrange  the  others  in  the  order  of  increasing  magni- 
tude. If  the  elements  are  letters  a,  b,  c,  d,  ••-,  we  naturally 
tliink  of  their  alphabetic  order  as^being  the  most  important. 

Whenever,  for  ayiy  reason,  one  of  the  n !  permutations  of  n 
symbols  is  to  be  regarded  as  more  important  than  any  other,  we 
call  it  the  principal  permutation. 

Let  X,  c,  m,  y,  b,  •••  be  tlie  principal  permutation  of  a  sys- 
tem of  n  symbols.  We  may  rename  x  and  call  it  aj,  rename 
c  and  call  it  a^,  and  so  on.  Thus,  by  changing  the  names  of 
the  elements,  if  necessary,  we  cati  always  fix  our  notation  in 
such  a  ruay  that  the  principal  permutation  will   assume  the 

form 

rtj  a^a^  •••  rt„ 

in  ivhich  the  subscripts  1,  2,  3,  •••,  n  appear  in  their  natural 
order. 

In  every  other  permutation  of  w  elements  a^,  •••,  a„,  some 
of  the  lower  subscripts  will  be  preceded  by  a  higher  one. 
Every  instance  of  this  kind  is  called  an  inversion.  By  the 
number  of  inversions  in  a  permutation  we  mean  the  total  number 
of  instances  in  ndiich  a  loiver  subscript  is  preceded,  in  that  per- 
mutation, by  a  higher  one. 

Thus,  in  the  permutation  23514G7  of  the  numbers  from  1  to  7,  there  are 
four  inversions.  1  is  preceded  by  2,  3,  5  (3  inversions).  Neither  2  nor  3  is 
preceded  by  a  higher  number.  4  is  preceded  by  5  (1  inversion).  5,  6,  7 
are  not  preceded  by  higher  numbers.    Thus  there  are  3  +  1  =  4  inversions. 

A  permutation  is  called  even  or  odd  according  as  it  contains 
an  even  or  an  odd  number  of  inversions. 


360  PERMUTATIONS   AND   COMBINATIONS      [Art.  218 

The  principal  permutation  contains  no  inversions  and  is  regarded  as 
an  even  permutation,  thus  leading  ns  to  classify  zero  as  an  even  number. 

Theorem  1.  If  any  two  elements  of  a  permutation  are  in- 
terchanged^ the  class  of  the  pefmutation  tvill  he  changed  from 
even  to  odd,  or  vice  versa. 

Proof.  Let  us  begin  with  the  case  in  wliich  a  pair  of  adjacent  elements 
are  intei-changed.  Let  these  adjacent  elements  be  called  a^  and  a^  and 
suppose  that /i<  ^•.  We  may  represent  by  a  single  letter  .1  the  collec- 
tion of  all  of  the  elements  of  the  permutation  which  precede  the  pair 
QhOki  and  by  B  the  collection  of  all  those  elements  which  follow  Offlif. 
If  the  original  permutation  was 

(1)  Aa^akB,     h  <  k, 

the  second  one,  after  the  interchange  of  o^  and  a^.,  will  be 

(2)  AakOhB,    h  <  A-. 

In  both  of  these  permutations  every  element  of  A,  and  also  every  ele- 
ment of  B,  is  preceded  by  exactly  the  same  elements,  and  therefore  by 
the  same  number  of  elements  with  higher  subscripts.  But  in  (2)  a^  is 
l^receded  by  a/c  (/;  <  k).  Therefore  (2)  contains  one  more  inversion 
than  (1). 

If,  instead,  Aaua^B  was  the  oi'iginal  permutation,  and  Aa^akB  the 
second  one,  we  see  by  the  same  argument  that  the  second  permutation 
contains  one  inversion  less  than  the  original  one.  In  either  case,  the 
number  of  inversions  is  changed  by  one,  and  therefore  the  class  of  the 
permutation  is  changed  from  even  to  odd  or  vice  versa. 

If  the  two  elements,  o^  and  Uk,  which  axe  to  be  interchanged  are  not  adja- 
cent, we  may  proceed  as  follows:  Let  there  be  m  elements,  ci,  C2,  ■■•,  r„„ 
between  a^  and  a^,  so  that  the  original  permutation  may  be  represented  by 

(;3)  Aa!,CiC2   •••   CmOkB, 

where  A  represents  the  collection  of  elements  which  precede  a^,  and  B 
the  collection  of  elements  which  follow  aa.-.  We  can  accomplish  the 
interchange  of  «/,  and  rtt  by  a  series  of  operations  each  of  which  con- 
sists in  merely  interchanging  adjacent  elements.  Thus,  we  obtain,  in 
order,  the  following  permutations  : 

AayfivCiCz  •••  c^-iCmakB,  the  original  permutation, 
Acia^^C2C:i  •••  Cm-iCmakB,  result  of  1st  operation, 
AcidtthCs  •■•  Cm-iCmCikB,  result  of  2d  operation. 


AciC2CsCi  •••  ('m-i'^'hCm.(ikB,  result  of  m  —  1th  operation, 
Aciczc^d  •••  c,n-\c„/iiflkB,  result  of  mth  operation. 
Acic-zCzCi  •••  Cm-ic^QkahB,  result  of  m  +  1th  operation. 


Art.  lMO]  COMBINATIONS  361 

After  these  m  +  1  operations,  Uh  actually  occupies  the  place  originally 
occupied  by  a,;.  But  a^t  is  not  yet  in  the  place  originally  occupied  by  o^. 
To  get  it  into  that  place,  we  change  the  last  of  the  above  permutations 
by  interchanging  a^  and  c„,  giving  the  new  permutation 

Ac^c.fiC^  •■•  QkC^ahB. 

As  a  result  of  m  operations  of  this  kind,  we  finally  obtain  the  desired 

permutation 

(4)  AokC^c.^  •••  c„ahB. 

Thus,  the  interchange  of  o/,  and  Uk  is  equivalent  to  (  »?+  \)  +  m  —  1  m 
+  1  interchanges  of  adjacent  elements.  Since  each  of  these  interchanges 
changes  the  class  of  the  permutation  from  even  to  odd,  or  vice  versa, 
and  since  2  m  +  1  is  an  odd  integer,  the  total  effect  of  interchanging  a^ 
and  flfc  will  be  to  change  the  class  of  the  permutation,  and  we  have 
proved  our  theorem.     We  may  also  state  this  same  theorem  as  follows: 

Theorem  2.  The  interchange  of  any  two  elements  of  a  per- 
mutation changes  the  number  of  inversions  by  an  odd  integer. 

Let  us  think  of  a  list  composed  of  all  of  the  n  !  permutations  of  n 
elements.  If  we  intei'change  any  two  of  the  elements,  say  the  first  and 
second,  in  every  one  of  these  permutations,  the  list,  after  this  change  has 
been  made,  will  still  include  all  of  the  permutations  but  not  in  the  origi- 
nal order.  For  two  permutations  different  from  each  other  before  the 
change  will  also  be  different  from  each  other  after  the  change.  But  by 
this  change  every  even  permutation  is  converted  into  an  odd  one  and 
vice  versa.  Hence  the  number  of  even  permutations  must  be  equal  to 
the  number  of  odd  permutations.     This  gives  the  following  theorem. 

Theorem  3.  Of  the  total  number  (n!)  of  permutations  of 
n  elements,  one  half  are  even  and  the  other  half  are  odd. 

These  theorems  are  of  great  importance  in  the  theory  of 
determinants  of  the  ni\\  order.     (See  Chapter  XII.) 

EXERCISE    XCIX 

Count  the  number  of  inversions  in  each  of  the  following  permuta- 
tions of  the  natural  numbers. 

1.    43-21.  2.    1324.  3.    .54321.  4.    13524. 

219.  Combinations.  If  n  elements  are  given,  we  may 
select  k  of  these  elements  in  viirious  ways.  Erery  set,  of  k 
elements  each,  which  can  be  obtained  from  n  elements,  no  atten- 


362  PERMUTATIONS   AND   COMBINATIONS     [Art.  219 

tion  being  given  to  the  order  in  which  they  may  be  arranged, 
is  called  a  combination  of  the  n  elements. 

Thus  out  of  three  elements,  a,  h,  c,  we  can  form  three  combinations  if 
we  take  two  at  a  time ;  namely  be,  ca,  and  ab.  The  combinations  be  and 
cb  are  the  same,  since  no  attention  is  to  be  paid  to  the  order  or  arrange- 
ment of  the  elements,  although  the  permutations  be  and  eb  are  different. 

We  have  denoted  by  ^Pk  the  number  of  permutations  of 
n  elements  takmg  k  of  them  at  a  time.  Let  us  denote  simi- 
larly by  J^C^.  the  number  of  combinations  of  n  elements  taken 
A;  at  a  time.  From  each  of  these  combinations  we  can  form 
h !  permutations  by  writing  the  k  elements  of  the  combina- 
tion in  their  k !  different  orders.     Consequently  we  have 

SO  that  we  find 

(1)  nOk  =  '^=  ''• 


k\       kl{n-k)l' 

if  we  make  use  of  formula  (2)  of  Art.  215.     This  formula 
may  also  be  written  as  follows 

,„.                 /7  _yt(n-l)(n-2)  ...  (n-k  +  1) 
C^;  n^k- ^^ ^, 

on  account  of  (1),  Art.  215. 

Whenever  we  select  k  elements  out  of  a  total  number  of 
n  elements,  there  are  n  —  k  left.  Consequently,  to  every 
combination  of  n  elements  taken  ^  at  a  time,  there  corre- 
sponds just  one  combination  of  the  n  elements  taken  n  —  k 
at  a  time.  There  must  exist,  therefore,  just  as  many  com- 
binations taken  ^  at  a  time,  as  there  are  combinations  taking 
n  —  k  at  a  time,  that  is,  it  must  be  so  that 

C")  n^k=^n^n~kl 

a  formula  which  may  also  be  verified  directly  by  means  of 
(1)  or  (2). 

Of  course,  we  have  in  particular 

(4)  nC-'i=„C„_i  =  7l, 

and 

(5)  „(7.  =  1, 


Art.  220]  INDEPENDENT   COMBINATIONS  363 

since  there  is  only  one  combination  of  n  things  taken  all 
together.  The  application  of  formula  (3)  in  this  case 
would  give 

and  we  shall  occasionally,  as  a  mere  matter  of  convenience, 
make  use  of  this  formula  although,  from  our  original  point 
of  view,  combinations  which  contain  no  element  were  not 
contemplated. 

In  deducing  formula  (1)  we  assumed  that  the  elements 
which  were  to  be  combined  were  all  different  from  each 
other.  The  formula,  of  course,  undergoes  a  modification 
if  some  of  the  elements  are  alike. 

220.  Independent  combinations.  The  following  problem 
in  combinations  arises  frequently.  There  are  «  classes  of 
elements  of  which 

the  1st  class  contains  a  elements, 
the  2d  class  contains  h  elements, 
the  3d  class  contains  c  elements, 
etc.         etc.  etc. 

We  are  to  form  combinations  of  s  elements  by  picking  out 
one  element  and  only  one  out  of  each  class.  In  how  many 
ways  may  this  be  done? 

We  select  any  one  of  the  elements  of  the  first  class,  any 
one  of  the  second  class,  any  one  of  the  third  class,  and  so  on. 
Clearly  there  are  i 

different  ways  of  doing  this. 

In  particular,  if  each  of  the  «  classes  contains  the  same 
number  a  of  elements,  the  number  of  combinations  is  a'. 

EXERCISE   C 

1.  Write  out  all  of  the  combinations  which  contain  two  of  the  five 
symbols  a,  b,  c,  d,  e. 

2.  Write  out  all  of  the  combinations  which  contain  three  of  the  five 
symbols  a,  h,  c,  d.  e,  and  state  a  reason  why  the  number  of  these  combi- 
nations is  the  same  as  of  those  found  in  Ex.  1. 


364  PERMUTATIONS   AND   COMBINATIONS      [Art.  221 

3.  If  four  points  in  the  same  plane,  no  three  of  which  ai'e  on  the 
same  straight  line,  are  joined  in  pairs  by  straight  lines,  how  many  of 
these  lines  will  there  be  ? 

Note.  The  configuration  obtained  in  this  way  is  called  a  complete 
quadrangle. 

4.  In  how  many  points  will  four  straight  lines  of  the  same  plane 
intersect? 

Note.  Four  lines  of  the  same  plane  together  with  all  of  their  points 
of  intersection  are  said  to  form  a  complete  quadrilateral. 

5.  How  many  lines  are  obtained  by  joining  n  points  of  the  same 
plane  by  straight  lines  in  all  possible  ways? 

6.  If  we  have  a  system  of  n  points  in  space,  such  that  no  four  of 
these  points  lie  in  the  same  plane,  how  many  planes  do  we  obtain  by 
passing  a  plane  through  each  set  of  three  of  the  n  points? 

7.  Prove  „C;,  =  „C„_i  by  using  (1)  or  (2)  of  Art.  219. 

8.  Find  the  values  of  ^^C^^  and  looQs- 

9.  How  m.any  committees  of  9  can  be  selected  from  a  group  of  12 
men  ? 

10.  A  committee  of  six  is  to  be  selected  from  seven  Englishmen  and 
four  Americans.  The  committee  is  to  contain  at  least  two  Americans. 
In  how  many  ways  may  the  committee  be  chosen  ? 

221.  The  binomial  theorem.  The  foinuila  (2)  of  Art.  219 
for  „(7^  is  precisely  the  same  as  the  expression  (2)  of  Art.  88 
for  tlie  coefficient  of  rc"~V'  in  the  expansion  of  (x  +  a)". 
Of  course  this  agreement  is  not  a  mere  accident.  In  fact, 
we  shall  now  give  a  new  proof  of  the  binomial  theorem  based 
on  the  theory  of  combinations. 

Let  us  consider  n  binomials  .r  +  f7i,  x  +  a-y,  ■■•.  x  -\-  a„  and  let  us  form 
their  product, 
(1)  (.r  +  (t\){x+  n.^  •••  (.r  +  «„)• 

If  this  product  is  multiplied  out  we  shall,  of  course,  obtain  an  integral 
rational  function  of  x,  of  degree  n.  The  complete  product  will  be  the 
sum  of  all  of  the  partial  products  which  can  be  obtained  by  selecting 
one  and  only  one  term  from  each  of  the  binomial  factors  of  (1),  and 
nniltiplying  these  together.  In  order  that  such  a  partial  product  may 
contain  exactly  the  (n  —  ^•)th  power  of  x  (no  higher  and  no  lower  power) 
as  a  factor,  n  —  k  of  the  factors  of  the  partial  product  must  be  x's,  and  the 
other  k  factors  must  be  a's.     Consequently,  there  will  be  as  many  partial 


Art.  222]      TOTAL   NUMBER  OF   COMBINATIONS  365 

products  of  tlii.s  kind  (containing  x"~'-"  as  a  factor)  as  there  are  ways  of 
selecting  L-  of  the  as  from  ai,  a^,  ••-,«„•  Tiierefore,  there  will  be  „Ca 
partial  products  iu  the  expansion  of  (1)  which  contain  x''~*  as  a  factor. 
Each  of  these  partial  products  will  reduce  to  x"~*<j*  if  ai.  (I2.  ••■,a„  are 
all  equated  to  a.  Since  there  are  „C*  such  partial  products  their  sum 
will  then  be  equal  to  „C4X"~*a*.  But  if  ai  =  «..  =  •••  —  «„  =  a,  the 
product  (1)  reduces  to  (x  +  a)",  and  therefore  we  find 

(2)  (x  +  a)"  -  x"  +  „(>"-i  4-  „C,a;"--  +  •••  +  „tV~*a*  +  ••• 

which  is  precisely  tlie  binomial  formula,  if  we  remember  that 
(A)  c  -^("  -  1)  •••  (»  -  A-+  1) 

222.  Total  number  of  combinations.  If  we  have  n  objects, 
we  may  form  combinations  of  them,  taken  one  at  a  time,  two 
at  a  time,  ••• ,  ^  at  a  time,  •••,  and  finally  all  of  them  together. 
The  total  number  of  combinations  will  be 

But  if,  in  equation  (2)  of  Art.  221,  we  put  2;  =  a  =  1,  we 

finrl 

(l  +  l)n=l  +  „C\  +  „C2+    •••    +na, 

SO  that  the  total  number  of  combinations  is  equal  to 
(1)  nO,+  nC,+  •■■  +„C„  =  2"-1. 

EXERCISE   CI 

1.  How  many  different  sums  of  money  may  be  formed  with  a  penny, 
a  nickel,  a  dime,  a  quarter,  a  half  dollar,  and  a  dollar? 

2.  A  merchant  has  a  set  of  12  different  weights.  IIow  many  differ- 
ent weights  can  be  obtained  from  these  by  combination? 

3.  In  how  many  ways  can  two  letters  be  filed  in  five  pigeonholes? 

4.  In  how  many  ways  may  52  cards  be  dealt  to  four  persons,  if  each 
person  plays  for  liimself  ?  (Only  those  ways  are  to  be  regarded  as  differ- 
ent which  have  some  influence  on  the  game.) 


CHAPTER   XI 

PROBABILITY 

223.  Definition  of  Probability.  As  our  knowledge  of 
Natui-e  becomes  deeper  and  more  extensive,  we  become  more 
and  more  convinced  that  tlie  laws  of  Nature  are  permanent 
and  universal  and  that,  in  the  strictest  sense  of  the  word, 
there  is  no  such  thing  as  chance.  Nevertheless  there  are 
many  phenomena  which  may  be  studied  from  the  point  of 
view  of  chance.  Thus,  if  we  toss  a  coin,  it  is  quite  evident 
that  the  peculiar  way  in  which  we  toss  it  will  cause  it  to 
land  either  head  or  tail  and  that  the  result,  whether  head  or 
tail,  is  really  determined  by  our  method  of  tossing  and  the 
laws  of  mechanics.  But  these  laws  are  difficult  to  apply  to 
any  specific  instance  of  this  sort,  and  a  very  slight  impercep- 
tible change  in  our  method  of  tossing  would  change  the 
result.  We  therefore  profess  complete  ignorance  of  the 
causes  which  govern  the  process  of  tossing  coins,  and  say 
that  the  two  possible  results,  head  or  tail,  are  equally  likely 
or  equally  probable,  and  that  the  probability  of  tossing  a 
head  is  equal  to  |.  Again,  if  a  bag  contains  three  white  and 
seven  black  balls,  and  if  we  draw  out  a  ball  at  random,  we 
say  that  the  probability  of  drawing  a  white  ball  is  -j^^,  since 
there  are  three  chances  out  of  ten  for  this. 

In  all  such  cases,  to  use  general  language,  we  estimate  the 
probability  of  an  event.  In  the  above  illustrations,  the 
event  is  either  the  appearance  of  a  head  after  the  coin  has 
been  tossed,  or  the  appearance  of  a  white  ball  as  a  result  of 
the  drawing.  By  a  trial  we  mean  any  operation  which  gives 
the  event  an  opportunity  to  happen.  In  our  illustrations 
the  tossing  of  the  coin,  or  the  act  of  drawing  a  ball  from  the 
bag,  are  trials. 

366 


Art.  223]  DEFINITION  OF  PROBABILITY  367 

In  estimating  the  probability/  of  an  event,  it  is  very  important, 
first  of  all,  to  decide  which  ones  of  the  various  possibilities  are 
to  be  regarded  as  equally  probable. 

Thus  in  tossing  a  coin  tliere  is  no  reason  to  suppose  that  one  side  will 
turn  up  rather  than  the  other.  Consequently  we  say  that  the  probability 
of  a  toss  resulting  head  is  ^,  and  the  probability  for  tail  is  also  |. 

But  suppose  we  aie  tossing  two  coins.  Either  face  of  either  coin  may 
turn  up,  giving  2  x  '2  =  i  possibilities.  We  might  be  led  to  conclude 
that  a  double  head  would  be  just  as  likely  as  a  combination  of  head  on 
one  coin  and  tail  on  the  other.  But  this  would  be  erroneous.  For  a 
double  head  is  only  one  of  four  possible  cases,  so  that  its  probability  is  |. 
But  there  are  two  chances  out  of  four  for  a  combination  head-tail,  since 
we  obtain  such  a  result  in  both  of  the  following  cases :  (1)  the  first  coin 
lands  head  and  the  second  tail,  (2)  the  second  coin  lands  head  and  the 
first  lands  tail.  Thus,  the  probability  of  tossing  head-tail  with  two  coins 
is  I  or  |. 

Suppose  we  have  made  a  list  of  all  of  the  results  which  mag 
possibly  apjjear  when  we  make  a  trial,  and  that  each  of  these 
results  is  as  likely  as  any  other.  Let  us  call  each  of  these 
possibilities  a  case,  and  let  there  be  m  +  n  cases  all  together. 
Suppose  that  m  of  these  cases  are  favorable  to  the  event  under 
consideration  ;  that  is,  let  ^ls  suppose  that  each  of  the  m  cases 
makes  it  certain  that  the  event  will  happen.  Let  us  suppose 
further  that  each  of  the  remaining  n  cases  is  unfavorable,  that 
is,  makes  it  certain  that  the  event  will  not  happen.  Then  ice 
say  that  the  probability  of  the  event  is 

.^ .  m 

the  ratio  of  the  favorable  to  all  possible  eases. 

The  probability  that  the  event  will  not  happen  is 


(2) 

p'  =^^1— 
m  +  n 

so  that 

(3) 

p  +  p'=l, 

(•i) 

p'=l-p. 

368  PROBABILITY  [Art.  224 

Therefore,  the  sum  of  the  probability  that  an  event  will  happen 
and  the  probability  that  it  will  fail  is  equal  to  unity. 

If  all  possible  cases  are  favorable,  then  w  =  0  so  that, 
according  to  (1),  we  have  p  =  1.  If  all  possible  cases  are 
unfavorable,  then  ^  =  0.  In  all  other  cases  p  is  a  positive 
proper  fraction.  Consequently  the  probability  of  an  event  is 
always  a  positive  proper  fraction  unless  we  are  certain  either 
that  the  event  will  happen  or  else  that  it  will  fail.  Tlie  prob- 
ability of  an  event  which  is  certain  to  happen  is  equal  to 
unity;  that  of  an  event  which  is  sure  to  fail  is  equal  to  zero. 

EXERCISE  CM 

1.  A  die  has  2  white  and  4  black  sides.  AVhat  is  the  probability  that 
a  white  side  will  turn  up  ? 

2.  Five  coius  are  tossed.  What  is  the  probability  that  the  result  will 
be  five  heads  ? 

3.  What  is  the  probability  of  tossing  4  heads  at  one  throw  with  5 
coins? 

4.  What  is  the  probability  of  tossing  3  heads  at  one  throw  with  5 
coins  ? 

5.  Show  that  if  there  are  n  coins,  the  probability  of  /•  heads  and  n  —  r 
tails  at  one  throw  is  nCr/'^'^- 

6.  An  ordinary  die  has  six  faces  marked  1.  2,  .'},  4,  .3,  6  respectively. 
What  is  the  probability  of  throwing  two  sixes  with  two  dice? 

7.  What  is  the  probability  of  a  score  of  11  with  two  dice? 

8.  If  three  dice  are  thrown,  what  are  the  probabilities  of  throwing 
three  sixes?  two  sixes  and  a  five?  a  six,  a  five,  and  a  four? 

224.  Compound  events.  It  is  frequently  convenient  to 
think  of  an  event  as  being  composed  of  several  simpler 
events. 

Thus,  if  we  are  tossing  two  coins,  we  may  i^refer  to  think  of  what  hap- 
l)ens  (head  or  tail)  when  we  toss  each  of  the  coins  separately,  and  speak 
of  the  result  of  tossing  the  two  coins  together  as  a  compound  event. 

But  whenever  we  decompose  any  event  whose  probability 
is  to  be  computed  into  simpler  events,  it  is  very  important 
to  know  wliether  the  component  simpler  events  are  independ- 
ent, dependent,  or  exclusive. 


Akt.  2-H]  COMPOUND   EVENTS  369 

Two  or  more  events  are  said  to  be  independent  or  dependent, 
aecording  as  the  occurrence  of  one  of  them  at  a  given  trial  does 
not  or  does  affect  the  prohahility  of  the  others. 

Suppose  we  are  drawiug  two  balls  out  of  a  bag  containing  three  white 
and  seven  black  balls.  We  may  decompose  this  process  into  two  separate 
drawings  of  one  ball  at  a  time.  The  probability  of  a  white  ball  at  the 
first  drawing  is  y*j.  But  if  we  have  actually  drawn  a  white  ball,  the 
probability  of  a  white  ball  at  the  second  drawing  will  be  |.  If  the 
first  ball  had  been  black,  the  probability  of  a  white  ball  at  the  second 
drawing  would  be  |.  Thus  the  probability  of  having  the  second  drawing 
result  in  a  white  ball  is  aifected  by  the  result  of  the  first  drawing. 

If  instead  tlie  balls  are  returned  to  the  bag  after  each  drawing,  the 
probability  of  a  white  ball  will  be  the  same  at  each  trial.  In  this  case, 
the  event  of  drawing  a  second  white  ball  would  be  independent  of  the 
event  of  drawing  a  first  white  ball. 

Two  or  more  events  are  said  to  he  exclusive  if  only  one  of 
them  can  happen. 

For  instance  the  two  events  of  a  single  coin  landing  head  and  landing 
tail  at  a  single  toss  are  exclusive.     Either  may  happen,  but  not  both. 

If  |)j,  p.^,  ••'-,  Pn  <^>'^  the  probabilities  of  n  independent  events, 
the  prohahility  that  all  of  these  events  will  happen  together,  at  a 
given  trial,  is 

(1)  P=PlP'i-'  Pn^ 

the  product  of  their  separate  probabilities. 

Proof.  Let  us  suppose  that  there  are  two  events  only.  Let  there  be 
rtj  cases  favorable  to  the  first  event  and  />j  which  are  unfavorable.  Then 
the  probability  of  the  first  event  is 


Pi  = 


ij_ 


Oj  +  i, 
Similarly  we  shall  have 

^2  =  — fr 

if  a„  cases  are  favorable  to  the  second  event,  and  ^2  are  unfavorable.  Since 
the  events  are  independent,  both  of  them  will  happen  together  in  a.^a.2^ 
cases  out  of  a  total  number  of  (ai  +  h\)(^a-i  +  b-i)  cases,  thus  giving 

, n^a„ 


370  PROBABILITY  [Art.  224 

as  the  probability  that  both  may  happen  together.  We  may  now  regard 
the  combination  of  these  two  events  as  a  single  event  of  a  probability  p'. 
The  probability  that  a  third  independent  event  of  probability /jg  may  also 
happen  will  therefore  be 

P'  Pz  =  PiP-.Pz- 

If  we  continue  to  reason  in  this  way,  we  finally  obtain  (1)  which  was  to 
be  proved. 

We  have  a  similar  method  for  calculating  the  probability  of  a  com- 
pound event  which  is  composed  of  several  dependent  events.  This 
method  may  be  expressed  by  means  of  the  same  formula  (1),  but  with  a 
somewhat  different  meaning  attached  to  the  symbols  pi,  p.j,  •••,  jo„,  as  ex- 
plained in  the  following  theorem. 

Let  p^  be  the  prohahility  of  a  first  event ;  let  p^  he  the  proba- 
bility of  a  second  event  after  the  first  has  happeiied  ;  let p^  be  the 
probability  of  the  third  event  after  the  first  two  have  happened  ; 
and  so  on.  Then  the  probability  that  all  of  these  events  will 
occur  together  is 

(2)  ^  =  PiP2Ps---Pn- 

Proof.  Let  us  suppose  first  that  there  are  two  events  only.  If  there 
are  a^  cases  favorable  and  b^  cases  unfavorable  to  the  first  event,  we  have 

«i 
pi- 


«i  +  ^1 

Now  both  events  cannot  happen  together  unless  the  first  event  happens. 
Consequently  the  cases  which  favor  both  events  can  be  found  only  among 
the  Oj  cases  which  favor  the  first.  If  62  of  these  cases  favor  the  second 
event,  the  probability  that  both  events  will  occur  is 

P'  = 


Oj  +  61 


If  the  first  event  has  actually  happened,  one  of  the  a^  cases  favorable  to  it 
must  have  occurred.  Since  b^  of  these  a^  cases  are  favorable  to  the  second 
event,  the  probability  of  the  latter,  after  the  first  has  occurred,  will  be 

But  we  have 

I  _       ^2     —      ^1       ^2  _ 

thus  proving  our  theorem  in  the  case  of  two  events.  The  extension  to  n 
events  is  made  as  in  the  proof  of  the  preceding  theorem. 


Art.  224]  COMPOUND  EVENTS  371 

The  following  theorem  is  concerned  with  mutually  exclu- 
sive events. 

Let  pi^  p^,  ••■,Pn  ^^  i^^  probabilities  of  n  mutually  exclusive 
events.  The  jj^'obabiHt//  that  one  of  these  events  (we  care  not 
which}  will  occur,  is  equal  to  the  sum  of  their  separate 
probabilities. 

Proof.  Let  there  be  m  equally  probable  cases.  Let  ai  of  these  be 
favorable  to  the  first  event,  a.,  to  the  second  event,  and  so  on.     Then 

a,  a„  (/„ 

■^'       to'  to  to 

Since  the  events  are  mutually  exclusive,  the  «,  cases  which  favor  the 
first  event,  the  a.^  cases  which  favor  the  second  event,  and  so  on,  must  be 
entirely  different  from  each  other.  Consequently  there  are  just 
«i  +  flo  +  •••  +  On  cases  among  the  m  eqiuiUy  jii-obable  cases  which  are  fa- 
vorable to  one  of  these  events,  and  the  probability  of  one  of  these  events  is 

a-,  +  a.y  +  ■■■  +  a,, 

'  '  -^=    P1  +  P2+    -    +Pn, 


711 

as  was  to  be  proved. 


EXERCISE    cm 


1.  A  traveler  has  five  connections  to  make  in  order  that  he  may  reach 
his  final  destination  on  time.  He  estimates  that,  for  each  of  these  con- 
nections, the  chances  are  two  to  one  in  his  favor.  What  is  the  probability 
of  his  getting  through  on  time? 

2.  A  traveler  argues  as  follows  before  starting  on  a  sea  voyage.  It  is 
an  even  chance  that  the  ship  will  encounter  a  storm.  The  probability 
that  she  will  spring  a  leak  in  tlie  storm  is  ^V-  If  ^  l*^^k  occurs  the  chances 
are  9  to  10  that  the  engine  will  pump  her  out.  Tf  they  fail,  the  chances 
are  3  to  4  that  the  compartments  will  keep  the  ship  afloat.  If  she  sinks, 
the  chances  are  even  for  him  to  be  saved  by  a  boat.  What  is  the  prob- 
ability of  his  being  lost  at  sea? 

3.  The  probability  that  A  will  live  ten  years  is  |,  and  the  probability 
that  B  will  live  ten  years  is  ^.  Wliat  is  the  probability  that  both  of 
them  will  be  alive  after  ten  years? 

4.  A  bag  contains  two  white,  three  black,  and  four  red  balls.  If  a 
ball  be  drawn  from  the  bag  at  random,  what  is  the  probability  that  it 
will  be  either  white  or  red  ? 

5.  What  is  the  prol)al)ility  of  throwing  either  an  ace  or  a  deuce  in  a 
single  throw  with  one  die  ? 


372  PROBABILITY  [Art.  225 

225.  Repeated  trials.  If  p  denotes  the  prohahilitij  that  an 
event  will  happen  on  a  single  trials  the  prohahility  that  the  event 
will  happen  exactly  r  times  in  n  trials  is  equal  to 

(1)  ^CrpX^-py-'-. 

Proof.  We  separate  the  proof  into  three  cases ;  r  =  0, 
r  =  1,  and  the  general  case  when  r  has  any  value. 

I.  It  r  —  0,  the  event  does  not  happen  at  all ;  it  must  fail  at  each  of 
the  n  trials.  If  p  is  the  probability  of  the  happening  of  the  event  at  any 
trial,  1—p  is  the  probability  of  its  failing  to  happen  (Art.  223). 
Therefore,  since  the  n  results  which  correspond  to  the  n  trials  are  inde- 
pendent, the  probability  of  failure  in  all  n  trials  is  (1  —  p)"  (Art.  224). 

II.  Let  r  =  1.  If  the  event  is  to  happen  once  and  only  once,  it  must 
happen  at  one  trial  and  fail  at  the  other  n  —  L  The  probability  of 
its  happening  at  a  specific  one  of  these  trials,  say  the  first,  and  failing  at 
the  other  n  —  1,  is  jt»(l  —  p)""^.  (See  Art.  224.)  But  since  there  is  an 
equal  opportunity  for  it  to  happen  at  each  of  the  n  trials,  we  find  the 
value 

np(l  —  p)"~^ 

for  the  probability  that  the  event  will  happen  just  once  in  n  trials. 

III.  The  general  case.  In  order  that  the  event  may  happen  exactly 
r  times,  it  must  happen  on  some  combination  of  r  trials  and  fail  on  the 
remaining  n  —  r  trials.  The  probability  that  the  event  may  happen  on 
any  specific  combination  of  r  trials  and  fail  on  the  remaining  n  —  r  is 

/)-(!  -  p)"--.     (See  Art.  224.) 

Since  there  are  „C,  combinations  of  r  trials  out  of  n,  the  probability  that 
the  event  shall  happen  exactly  r  times  in  n  trials  is 

as  stated  in  the  theorem. 

Of  course,  on  account  of  (3)  of  Art.  219,  this  is  also  equal  to 

„C„_j;-(l -/))"-. 

In  many  cases  we  are  interested  in  the  question  whether  the  event 
occurs  at  least  r  times  in  n  trials  rather  than  whether  it  occurs  exartly  r 
times  in  n  trials.  Such  is  the  case,  for  instance,  if  a  chess  phiyer  under- 
takes to  win  three  games  out  of  four.  He  will  make  good  his  intention 
if  he  wins  either  three  or  four  games. 

The  solution  of  the  question  is  immediate.  The  event  will  happen  at 
least  r  times,  if  it  happens  exactly  n  times,  or  n  —  1  times,  •■•  ,  or  r  +  1 
times,  or  r  times.  Therefore  (see  Art.  224)  the  prnfxiJiillti/  Ihal  nn  ereiil 
of  prohahilitii  p  will  happen  (it  least  r  times  in  n  trials  Is 

(2)  p'^  +  r,c..^p--\\  -p)  +/Voi»"-'(i  -py+  -  +  uC\p\i-py. 


Art.  226]        APPLICATION   TO   LIFE  INSURANCE  373 

EXERCISE    CIV 

1.  Prove  by  the  methods  of  Article  225  the  result  of  Example  5, 
Exercise  OIL 

2.  What  is  the  probability  that  in  six  throws  of  a  coin,  at  least  three 
will  result  in  heads  ? 

3.  A  is  playing  chess  with  B.  Ilis  chances  of  winning  any  one  game 
from  B  are  2  to  1.  What  is  the  probability  that  A  will  win  three  games 
out  of  four? 

226.  Application  to  life  insurance.  When  a  company 
insures  a  man's  life  it  makes  a  contract  with  him  according 
to  which  the  company  agrees  to  pay  his  heirs,  after  proof  of 
death  has  been  offered,  a  certain  sum  of  money,  in  return  for 
payments  made  by  him  to  the  company  at  certain  stated 
intervals  during  his  lifetime.  These  payments,  called 
premiums,  may  be  continued  during  the  whole  life  of  the 
insured  person,  or  else  for  a  stated  number  of  years.  The 
company  undertakes  a  risk  in  each  individual  case,  since 
the  insured  may  die  long  before  his  premiums  are  sufficient 
to  cover  the  sum  for  which  the  company  is  liable  to  his 
heirs.  To  put  the  insurance  business  on  a  sound  basis  it 
is  necessary,  therefore,  to  know  what  is  the  probability 
that  a  person  of  a  certain  age  shall  live  a  certain  number 
of  years. 

In  order  to  be  able  to  do  this,  actuaries  have  constructed 
so-called  Tables  of  Mortality  which  are  based  on  statistics. 
Suppose  we  make  a  record  of  100,000  people  at  the  age  of 
25  and  find  that,  at  the  end  of  a  year  99,194  of  them  are  still 
living.  We  should  conclude  from  these  data  that  the  proba- 
bility of  surviving  one  year,  for  a  person  25  years  of  age, 
would  be  99,194/100,000  =  0.992,  The  results  obtained  in 
this  way  become  more  reliable  the  greater  the  numlier  of 
cases  which  have  been  taken  into  account.  The  Appendix 
contains  such  a  table  of  mortality. 

The  general  i)rinciple,  upon  which  this  method  of  esti- 
mating probabilities  is  founded,  may  be  formulated  as 
follows  : 


374  PROBABILITY  [Art.  226 

If  n  is  a  very  large  number  and  if  observations  show  that  a 
certain  event  has  happened  m  times  in  n  possible  cases,  the 
probability  of  this  event  is  equated  to  m/n.  In  other  words, 
the  assumption  is  made  that  at  the  next  opportunity  the  event 
will  again  happe7i  m  times  out  of  a  possible  n. 

To  find  the  probability  p  that  a  person  of  age  a  will  live  to 
age  5,  we  take  from  the  table  the  number  of  persons  living 
at  age  b  and  divide  it  by  the  number  living  at  age  a. 

If  the  insured  person  of  age  a  promises  to  pay  $  m  per 
year  until  he  is  b  years  old,  the  value  of  this  promise  to  the 
company  may  be  calculated  as  follows :  His  first  payment 
is  certain,  since  the  policy  will  not  be  in  force  until  the  first 
payment  is  made.  The  second  payment  is  contingent  upon 
his  living  at  least  one  year.  Let  p^  be  the  probability  for 
this,  as  obtained  from  the  table  of  mortality.  Then  the 
value  of  the  promise  to  pay  I  m  at  the  end  of  the  first  year, 
or  the  value  of  the  company's  expectation  is  estimated  to  be 
$7wpj.  Similarly  the  value  of  the  company's  expectation 
for  the  third  premium  of  I  m  is  f  mp^i  if  P2  ^^  ^^^®  probability 
that  the  insured  will  survive  two  years.  These  estimates 
are  in  accordance  with  the  principle  that  the  value  of  a  prob- 
able payment  (or  the  expectation)  is  equal  to  the  sum  to  be 
paid  midtiplied  by  the  probability  that  it  will  be  paid. 

Of  course,  in  the  case  of  life  insurance  the  actual  value, 
to  the  company,  of  each  payment  is  really  greater  than  in- 
dicated above.  For  the  money  is  invested  by  the  company 
and  produces  income.  Therefore  any  complete  solution  of 
the  question  of  the  value  of  these  payments  to  the  company 
must  take  into  account  the  interest  earned  by  each  payment. 
The  company  must  also  make  the  proper  deduction  for  the 
expenses  connected  with  the  conduct  of  its  business  and  for 
a  reasonable  profit.* 

Only  by  investigations  of  the  sort  here  indicated  can  the 
proper  rates  and  the  proper  methods  of  conducting  the  busi- 

*  For  a  more  detailed  discussion  of  life  insurance  and  other  applications  of 
algebra  to  commercial  questions,  see  E.  B.  Skinner,  T/>e  Mathematical  Theory 
0/  Investment,  Giun  &  Co.,  l'J13. 


Art.  227]  OTHER   APPLICATIONS  375 

ness  of  life  insurance  be  found.     Similar  questions  arise,  of 
course,  in  accident  and  fire  insurance. 

227.  Other  applications  of  the  theory  of  probability.  There 
are  many  other  applications  of  the  theory  of  probability. 
One  of  the  most  notable  of  these  is  the  theory  of  errors  of 
observation.'"'  Every  measurement  made  by  man  is  subject 
to  inaccuracies  due  in  part  to  unavoidable  defects  in  the  in- 
struments, and  in  part  to  physiological  causes.  Thus,  if  the 
same  quantity  is  measured  independently  several  times  in 
succession,  the  results  obtained  will  usually  differ  to  some 
extent.  The  questions,  how  to  obtain  the  most  probable 
value  from  these  discordant  results,  and  how  to  find  the 
probable  error  of  this  most  probable  result,  are  of  great 
importance  in  physics  and  astronomy. 

*  Consult  a  very  readable  presentation  of  this  important  subject  by  L.  D. 
Weld,  Theory  of  Errors  and  Least  Squares,  New  York,  The  MacmillauCo.,  191G. 


CHAPTER    XII 

DETERMINANTS    OF    THE    »iti>   ORDER   AND    SYSTEMS    OF 
LINEAR   EQUATIONS   "WITH   n   UNKNOWNS 

228.   Definition  of  a  determinant  of  the  nth  order.      We 

studied    determinants    of    the    second    and   third    order   in 
Cliapter  IX.     They  were  defined  by  the  equations 

=  a^  -  a,^h^  (Art.  202). 


(1) 

and 

«1 

«2 

^1 

«i    ^1 

H 

(2) 

«2    h 

H 

«3 

*3 

H 

=  «1^2^3  +  «2*3^1  +  ^3^1^2  -  «1^3^2  "  ^2*1^3  "  ^3*2^1 

(Art.  206). 

We  shall  now  make  use  of  these  expressions  to  suggest  by 
analogy  the  definition  for  a  determinant  of  the  nWi  order. 
In  both  (1)  and  (2)  we  have  a  square  array  of  numbers. 
The  first  term  in  the  right  member  is,  in  both  cases,  the 
product  of  all  of  the  elements  which  occur  in  that  diagonal 
of  the  square  array  which  passes  from  its  upper  left-hand  to 
its  lower  right-hand  corner.  All  of  the  other  terms  in  the 
right  member  are  obtained  from  this  first  or  principal  term 
by  making  a  permutation  of  the  subscripts  while  the  letters 
are  left  in  their  alphabetic  order.  Those  terms  are  preceded 
by  a  'plus  sign  whose  subscripts  are  so  arranged  as  to  form 
an  even  permutation,  the  principal  permutation  being  1,  2,  8. 
(See  Art.  218.)  The  other  terms  are  preceded  by  a  minus 
sign. 

Thus  13  2  has  one  inversion  (3  before  2).  It  is  therefore  an  odd 
permutation  and  the  corresponding  term  a\h;\c-i  of  (2)  is  preceded  by  a 
minus  sign.  Again  3  1  2  has  two  inversions  (3  before  1,  and  3  before  2). 
The  corresponding  term  a-^\C.^  is  preceded  by  a  plus  sign. 

370 


Art.  228]         DEFINITION   OK    A    DETKllMIXANT  377 

We  now  proceed  to  dt'liiic  a  dL-tenninant  of  the  nth 
order. 

1.  Consider  a  >(quare  arrai/  of  n^  numbers^  each  of  ivhich  is 
called  an  element  of  (he  determinant ;  this  array  may  he.  writ- 
ten as  follows : 


(3) 


^1'       1"'     ^r      "■■'       1' 

ag,        Oj'        ^2''        ■"'        ^2'' 


<fn^        ^V        'V         •••i        ^n. 

and  consists  of  n  (horizontal)  rows  and  n  {vertical)  columns. 
The  notation  has  been  chosen  in  such  a  way  that  the  eleme)its 
in  one  and  the  same  column  are  represented  hy  the  same  letter 
{column  mark)  while  the  elements  of  the  same  row  have  the 
same  i<uhxcript  {row  mark). 

'1.  The  diagonal  which  passes  from  the  upper  left-hand  to 
the  loirer  right-hand  corner  of  this  square  array  is  called  the 
principal  diagonal  and  the  jyroduct  of  the  n  numbers  in  this 
diagonal 

(4)  ajVg  ...  ^„ 

is  called  the  principal  term  of  the  determinant.  . 

3.  Let  us  consider  all  of  the  possible  products  which  can  be 
formed  from  the  array  (3)  by  taking  as  factors  one  elemeiit, 
and  only  one,  from  every  column  and  every  row.  One  of  these 
products  will  he  the  principal  term  (4).  Each  of  these  prod- 
ucts may  be  ivritten  in  the  form 

where  every  letter  {or  columti  mark)  occurs  once  and  only  o)ice, 
since  one  and  only  one  factor  is  to  be  taken  from  each  column, 
and  where  the  letters  a^e  ivritten  in  their  normal  order.  Among 
the  subscripts  ?'j,  i^,  •••,  V  &very  one  of  the  numbers  1,  2,  ..-,  w 
ivill  occur  once  and  only  once,  since  one  and  only  one  factor  of 

(5)  is  to  be  taken  from  each  row  of  (3). 

4.  Let  the  product  (5)  he  preceded  hy  the  plus  or  minus 
sign  according  as  the  permutation  ?'j,  i^,  i^,  •••,  i^  is  even  or  odd. 
the  permutation  1,  2,  •.-,  n  which  corresponds  to  the  principal 


378 


DETERMIXANTS   OF   THE   71^^   ORDER      [Art.  229 


term  being  regarded  as  the  principal  permutation.     (See  Art. 
218.) 

5.  There  are  n!  such  products.  (See  Art.  214.)  They 
are  called  the  terms  of  the  determinant.  The  algebraic  sum 
of  these  n  !  terms.,  the  sign  of  each  term  being  determined  by  the 
rule  given  in  No.  4,  is  a  number  called  the  determinant  of  the 
r?  numbers  (3). 

Thus  we  see  that  a  determinant  of  the  ?ith  order  is  a 
single  number  which  is  obtained  from  n^  given  numbers, 
arranged  in  a  square  arrajs  by  performing  the  various  opera- 
tions described  in  the  above  definition. 

The  symbol  for  such  a  determinant  is  obtained  from  the 
square  array  of  its  elements  (8)  by  inclosing  it  between  two 
vertical  lines.     Thus,  the  symbol 


(6) 


Z)  = 


b. 


k. 


h. 


calls  for  the  formation  of  a  single  number  D  from  the  n^ 

given  ones  by  means  of  the  operations  just  described. 

Another  notation,  in  some  respects  preferable  to  the  one 

used  above,  for  the  elements  of  a  determinant  is  shown  in 

the  formula 

fjj     aj2     ••.     a^^ 

ttfyt  ttnn  •  •  •  i'n 


(7) 


D  = 


"21 


22 


'■2n 


%l 


'«2 


in  which  each  element  has  two  subscripts,  the  first  subscript 
indicating  the  row  and  the  second  the  column  in  which  it 
occurs  in  the  array. 

229.  Another  method  for  determining  the  sign  of  a  term  of 
the  determinant.  In  our  definition  (Art.  228)  we  thought 
of  any  term  of  the  determinant  as  written  in  the  form 

(1)  ^i,K<^H    ■■■    ^in^ 


Art.  229]  SIGN  OF   ANY   TERM  379 

where  the  letters  a,  b,  ••',  k  were  in  their  alphabetic  order. 
The  sign  to  be  i)retixe(l  to  this  term  was  +  or  —  according 
as  ^'^,  i^-,  •••,  in  ^^'^^  an  even  or  odd  permutation  of  1,  2,  •••,  n. 
We  may,  however,  rearrange  the  factors  of  (1)  so  as  to 
make  the  subscripts  appear  in  the  natural  order  1,  2,  •••,  n. 
If  we  do  this,  the  letters  will  now  no  longer  appear  in  their 
alphabetic  order.  Let  us  count  tlie  number  of  inversions 
in  this  permutation  of  the  letters,  tliinking  of  the  alphabetic 
order  as  the  principal  permutation  (^see  Art.  218),  thus 
enabling  us  to  decide  whether  this  permutation  of  the  let- 
ters is  even  or  odd. 

We  shall  find  that  this  permutation  of  the  letters  is  even  or 
odd  according  as  the  permutation  of  the  subscripts  in  (1)  was 
even  or  odd. 

That  this  is  true  follows  from  the  following  remark.  In 
the  product  formed  from  the  factors  a^^^  b^,^,"-,  ki^,  taken  in 
any  specified  order,  let  us  interchange  any  two  of  the 
factors.  We  are  then  interchanging,  in  the  first  place,  two 
of  the  subscripts  and  therefore  we  are  changing  the  number 
of  inversions  of  the  subscripts  by  an  odd  number  (The- 
orem 2,  Art.  218).  At  the  same  time  we  are  interchanging 
two  of  the  letters,  so  that  the  number  of  inversions  occur- 
ring among  the  letters  will  also  be  changed  by  an  odd  num- 
ber. Let  us  add  the  number  of  inversions  of  the  subscripts 
to  the  number  of  inversions  in  the  letters.  The  change  pro- 
duced in  this  sum,  by  interchanging  two  of  the  factors  of 
the  product,  will  be  an  even  number,  since  the  sum  of  two 
odd  numbers  is  always  even.  Consequently,  the  sum  of  the 
number  of  inversions  of  the  letters  and  of  the  number  of  inver- 
sions in  the  subscripts  in  such  a  product,  will  not  change  its 
character  of  evenness  or  oddness  by  any  rearrangement  of  the 
factors. 

In  (1)  the  letters  are  written  in  their  natural  order,  that 
is,  with  no  inversions.  Let  there  be  i  inversions  among  the 
subscripts.  Rewrite  (1)  so  as  to  make  the  subscripts  occur 
in  their  natural  order,  that  is,  with  no  inversions,  and  let 


380 


DETERMINANTS   OF   THE   w^''   ORDER      [Art.  230 


j  be  the  number  of  resulting  inversions  among  the  letters. 
Then,  the  sums  0  +  ^  and  j  +  0  are  either  both  even  or  both 
odd,  that  is,  i  'dndj  are  both  even  or  botli  odd. 

Consequently,  we  may  determine  the  sign  of  any  term  of 
the,  determinant  hy  the  rule  of  No.  4  of  the  definition  given  in 
Art.  '22H.  We  inay  equally  well  rewrite  the  term  in  such  a 
way  as  to  cause  the  subscripts  to  appear  in  their  natural  order, 
and  prefix  the  plus  or  minus  sign  according  as  the  resulting 
permutation  of  the  letters  is  even  or  odd.  Both  methods  ivdl 
give  the  same  result.  , 

230.  Properties  of  determinants.  We  are  now  ready  to 
show  that  determinants  of  the  nt\\  order  have  essentially  the 
same  properties  which  we  found  in  Art.  208  for  determi- 
nants of  the  third  order. 

1.  A  determinant  of  the  nth  order  does  not  cha^ige  its  value 
if  its  elements  are  transposed,  that  is,  if  its  roivs  he  converted 
into  columns  and  its  columns  into  rows,  the  relative  order  of 
rows  and  columns  not  being  changed. 

Proof.     Let 


Z>  = 


We  wish  to  show  that 


ai 

h 

Cl      • 

•      ^1 

ao 

h 

Co       ■ 

•     k. 

a„ 

^n 

r„     • 

■    K 

D' 


Oi 

02 

as     ■ 

•       «n 

^'l 

h-2 

h     ■ 

•         f'n 

^-1 

h 

h    ■ 

■    K 

Z>=  D'. 

We  observe  in  the  first  place  that,  when  we  apply  the  definition  of 
Art.  228  to  each  of  these  determinants,  exactly  the  same  terms  will 
appear  in  both,  since  these  terms  are  formed  by  computing  all  possible 
products  of  n  elements  obtained  by  taking  one  and  only  one  element  from 
each  row,  and  one  and  only  one  element  from  each  column.  ]Moreover, 
the  corresponding  terms  of  the  two  determinants  will  have  the  same  sign 
prefixed  to  them,  on  accoimt  of  the  final  theorem  of  Art.  229.  Therefore 
the  two  determinants  are  equal. 

2.    If  all  of  the  elements  of  a  row  or  column  of  a  determi- 
nant are  equal  to  zero,  the  value  of  the  determinant  is  zero. 

For  every  term  of  the  determinant  will  contain  as  one  of  its  factors  an  ele- 
ment of  the  row  or  column  in  question,  and  will  therefore  be  equal  to  zero. 


Art.  230]  PROPERTIES  OF   DETERMINANTS  381 

3.  If  two  roivs  or  columns  of  a  determinant  are  interchanged^ 
the  relative  order  of  the  elements  in  such  rows  or  columns  not 
being  altered^  the  determinant  merely  changes  its  sign. 

Proof.  Change  of  two  rows  is  equivalent  to  an  interchange  of  two 
of  the  subscripts  (row  marks)  in  every  term  of  the  determinant.  But 
such  an  interchange  increases  or  diminishes  the  number  of  inversions 
among  the  subscripts  by  an  odd  number  (Theorem  2,  Art.  218),  and 
therefore  clianges  the  sign  of  tlie  corresponding  term  of  the  determinant. 
(No.  4,  Art.  228.)  Similarly  for  interchange  of  two  columns  according 
to  tlie  final  theorem  of  Art.  229. 

4.  If  all  of  the  elements  of  a  row  (or  column)  are  multiplied 
by  the  same  factor  w,  the  determinant  is  multiplied  by  m. 

Proof.  Of  the  elements  which  are  multiplied  by  m,  one  and  only 
one  occurs  in  every  term  of  the  determinant. 

5.  A  determinant  of  the  nth  order  is  equal  to  zero  if  two  of 
its  parallel  lines  (rows  or  columns)  read  alike,  that  is,  if  all 
pairs  of  corresponding  elements  in  two  parallel  lines  are  equal 
to  each  other. 

The  proof  is  exactly  the  same  as  for  determinants  of  the  third  order. 
(See  Theorem  5,  Art.  208.) 

6.  If  every  element  of  any  row  or  column  is  expressed  as  a 
sum  of  two  terms,  the  determinant  may  be  expressed  as  a  sum 
of  two  others. 

For  instance, 


"l  +  '"l» 

h, 

..     ki 

(lu 

bi, 

■■        h 

mi, 

bu 

•■      h 

do  +   1112, 

h, 

■  ■     hi 

= 

(!■>, 

b,, 

..        h 

+ 

'«2, 

b,, 

■■        1-2 

«„  +  m,., 

K, 

■■         f^-n 

««. 

bn, 

■■    K 

m„. 

/>„, 

•    K 

To  prove  this  theorem  observe  that,  as  a  result  of  the  hypothesis,  any 
term  of  the  original  determinant  may  be  expressed  as  a  sum  of  two  terms, 
each  of  which  is  one  of  the  terms  occurring  in  the  expansion  of  another 
determinant. 

7.  If  to  the  elements  of  any  row  or  column  we  add  the  corre- 
.^ponding  elements  of  a  parallel  row  or  column,  multiplied  by  one 
and  the  same  factor,  the  value  of  the  determinant  is  not  changed. 


382  DETERMINANTS   OF   THE   n**^   ORDER       [Art.  281 

Proof  as  in  the  case  of  third  order  determinants.  (See  Theorem  7, 
Art.  208.) 

Before  we  can  generalize  the  remaining  theorems  of 
Art.  208,  we  must  discuss  the  notions  of  minors  and  cofactors 
of  a  determinant  of  the  wth  order. 

231.  Minors.  Consider  a  determinant  of  the  nth  order  and 
suppress  both  the  row  and  the  column  in  which  any  particular 
element  lies.  The  determinant  of  the  {n  —  1)^^  order  formed 
from,  the  remaining  elements,  ivithout  disturbing  their  relative 
position,  is  called  the  minor  of  the  element  in  question. 

For  examples  of  such  minors  see  Art.  206. 

Let  us  use  the  notation  Dg  for  tlie  minor  of  any  element 
of  D  which  is  named  e.  Thus,  the  minor  of  «j  will  be  called 
Da^i  that  of  Cg  will  be  i><,^,  and  so  on.  We  then  find  the  fol- 
lowing theorem. 

1.  If  the  determinant  D  is  expanded,  the  sum  of  all  of  those 
terms  which  involve  the  element  a^  (^which  stands  in  the  upper 
left-hand  corner  of  D)  is  a^D^^. 

Proof.  According  to  the  definition  of  a  determinant  of  the  nth  order^ 
Art.  228,  every  term  of  D  which  contains  a^  as  a  factor  is  formed  by 
multiplying  a^  by  a  product  of  n  —  1  elements  chosen  from  the  n  —  1 
rows  and  columns  of  D  which  are  different  from  the  first  row  and  the 
first  column,  and  chosen  in  such  a  way  that  one  and  only  one  element  is 
selected  from  each  of  these  rows  and  columns.  Consequently  every  term 
of  this  kind  is  obtained  by  multiplying  a^  by  a  term  of  D^j.  Moreover, 
the  sign  which  precedes  such  a  term  of  D  will  be  the  same  as  that  which 
precedes  the  corresponding  terms  of  D^^,  since  the  number  of  inversions 
among  the  subscripts  in  any  term  of  D^^,  such  as 

'2     '3  '/l' 

will  not  be  changed  if  we  write  Oj  in  front  of  it. 

2.  If  we  denote  by  e  the  element  in  the  ith  roiv  and  the  kth 
column  of  D,  the  sum  of  all  of  the  terms  of  I),  lohich  have  e  as 
a  factor,  is  ^_  -^y+,^j)^^ 

To  prove  this  theorem,  we  observe  that  e  may  be  brought  into  the 
position  originally  occupied  by  a^  without  disturbing  the  relative  posi- 


Art.  231]  MINORS  383 

tion  of  any  of  the  elements  of  D  excepting  those  which  lie  in  the  row 
and  column  in  w^hich  e  stands.  This  may  be  done  by  first  interchanging 
the  row  in  which  e  stands  with  the  preceding  row  and  repeating  this 
operation  until  e  stands  in  the  first  row.  Since  e  was  originally  in  the 
ith  row,  i  —  \  such  operations  will  accomplish  this  result.  We  may  then, 
by  interchanging  columns  in  the  same  way,  bring  e  finally  into  the  posi- 
tion originally  occupied  by  Oj.  This  will  be  accomplished  after  k  —  \ 
further  operations,  making 

i-\  +  k-\  =  i+k-'2 

operations  in  all.  Each  of  these  operations  changes  tlie  sign  of  the 
determinant  (Theorem  3,  Art.  230).     Consequently  we  shall  have 

if  we  denote  by  D'  the  new  determinant  obtained  from  D  by  these 
i  -\-  k  —  2  operations.  The  minor,  D'^  of  e  in  Z>',  is  the  same  as  the 
minor,  D^  of  e  in  D,  since  the  elements  in  both  determinants  are  the  same, 
and  since  the  rows  and  columns  of  the  two  determinants,  excepting  only 
the  row  and  the  column  which  contains  e,  are  arranged  in  the  same  rela- 
tive order.     Hence  D^  =  D\. 

By  Theorem  1  the  sura  of  all  of  those  terms  of  D'  which  contain  e  as 
a  factor  is  eD'^  =  eD^'  Therefore  the  sum  of  the  corresponding  terms 
in  D  is 

as  we  wish  to  prove. 

If  this  proof  does  not  seem  quite  clear  to  the  student,  let  him  actually 
trace  the  changes  of  sign  which  take  place  as  some  element,  say  d^,  of 
the  determinant  of  the  fourth  order 

a^     a^     ag     a^ 

&i       &2       ^3       ^'4 

c-y     Co     <'^     c^ 
f/j     d.^     d^    d^ 

is  gradually  brought  into  the  upper  left-hand  corner. 

As  a  result  of  Theorem  2  we  obtain  at  once  the  following 
theorem. 

3.  Let  us  fix  our  attention  upon  any  particular  row  or 
column  of  a  determinant.  Multiply  every  element  of  such  a 
row  or  column  by  its  minor  and  prefix  the  plus  or  minus  sign 


384  DETERMINANTS  OF   THE   n^^   ORDER       [Art.  2:52 

to  this  pi'oduct  according  as  we  find  a  plus  or  minus  sign  in  the 
place  which  corresponds  to  that  element  in  the  diagram  of  signs 

+  -  +  -  • 
-  +  -  +  • 
+      -      +      -      • 


which  contains  n^  signs  arranged  in  a  square  array ^  alternating 
in  checkerboard  fashion^  the  signs  along  the  principal  diagonal 
all  being  -\- .  The  algebraic  sum  of  all  such  products,  with  the 
proper  sign  prefixed^  will  be  equal  to  the  value  of  the  deter- 
7ninant. 

This  follows  from  Theorem  2  because  (—  1)»+*  is  equal  to  +  1  or  —  1 
according  to  the  indications  given  by  the  diagram  of  signs. 

This  method  of  expressing  a  determinant  is  called  expand- 
ing with  respect  to  the  elements  of  a  definite  row  or  column. 
Since  the  minors  are  determinants  of  a  lower  order,  to  which 
the  same  method  of  expansion  may  be  applied,  this  method 
is  often  convenient  when  we  wish  to  calculate  the  value  of  a 
determinant.  (Compare  Art.  206  for  the  special  case  of 
determinants  of  the  third  order.) 

232.  Cofactors.  If  e  denotes  again  that  element  of  D  which 
stands  in  the  ith  row  and  the  kth  column,  the  quantitg 

which  differs  from  the  minor  Dg  at  most  in  sign^  is  called  the 
cofactor  of  e. 

Let  us  denote  by  Ai,  Bi,  •••,  A2,  Bo,  •••,  the  cofactors  of  di,  /y,,  ■••,  112,  h-i, 
•  ••  respectively,  in  tlie  deterniiiiaiit  D.     Then  we  may  write 


Z>=  r/,.li  4  (uAi  +  •••  +  a„A, 
D^h.B^  +  b.B.    +  •••  +  ^/J„ 

D=  f.Jx\  +  k-.K;  +  •••  +  KK,, 
also 


CI)  D^h.B,  +  b.B.    +  •••  +  hjl,. 


and  also 

D  =  niAy  +  h,Bi   +  •••  +  k\Ku 

D  =  a.A.  +  h,Bn   +  •■•  +  l.J\2, 


D  =a„.4„  +  IkB,^  +  •••  +  A„/v„. 


Akt.  232] 


COFACTORS 


385 


To  prove,  for  instance,  the  first  equation  of  (1)  it  suffices  to  remark 
that  flj^i  represents  the  aggregate  of  all  of  those  terms  of  D  which  con- 
tain Oj,  that  a.,.!,  is  equal  to  the  sum  of  all  of  those  terms  of  D  which 
contain  (z.^,  and  so  on. 

But  equations  (1)  and  (2)  are  equivalent  to  Theorem  8  of  Art.  208, 
which  was  there  proved  only  for  determinants  of  the  third  order. 
Theorem  9  of  Art.  208  may  be  generalized  to  determinants  of  the  rjth 
order  by  an  argument  whi(Oi  involves  merely  a  repetition  of  the  essential 
features  of  the  proof  given  in  Art.  208  for  third  order  determinants. 
Thus  we  have  the  further  result: 

Theorems  8  and  9  of  Art.  208  are  true  of  determinants  of 
any  order. 

EXERCISE    CV 

Compute  the  value  of  the  following  determinants  by  expanding  with 
respect  to  the  elements  of  some  row  or  column : 

1  2 

3  4 

1  1 

2  1 


1 

1 

o 

1 

0 

0 

3 

_  X 

3 

o 

1 

0 

1 

-  1 

1 

1 

2 

1 

2 

1 

3 

3 

1 

4 

4 

4 

3 

2 

5 

5 

2 

1 

3. 


0 

1 

-1 

2 

2 

1 

3 

1 

By  applying  some  of  the  theorems  about  determinants  evaluate  the 
following : 


a 

- 

a 

- 

a 

- 

a 

1 

1 

1 

a 

1 

1 

1 

a 

1 

1 

1 

b    c 
h    d 

h     c 

1 
1 
1 


1 

o 

rt2 

1 

h 

6^ 

1 

c 

C2 

1 

d 

rf2 

68 


Hint.     Compare  Ex.  5  in  Exer- 
cise xcni. 


la  +  mp,     11)  +  Hi  7,     Ic  +  mr,     Id  +  ms 


8.    Slaow  that 


a     h 

f  f 

0  0 
0     0     / 


c     d 

fl  '> 

a    b 

J     ^^ 

J  '^- 

«   / 

I     m 

I     m 

386 


DETERMINANTS   OF   THE   71^^   ORDER       [Art.  233 


233.  Solution  of  a  system  of  n  linear  equations  for  n  un- 
knowns. It  is  now  an  easy  matter  to  solve  n  linear  equa- 
tions, such  as 

a^i  +  ^i-J'a  +  •••  +  ^"i^\  =  ^p 

ag^i  -h  b^x^  +  ••  •  +  Vn  =  ^2' 

for  the  w  unknown  quantities  x^,  .j-g,  •••  a;„. 
Let  us  denote  bv  D  the  determinant 


(2) 


£> 


«1 

^     • 

■     '^i 

Qg 

h     ■ 

••     ^2 

a„ 

b„     ■ 

••       l^-n 

of  the  coefficients  of  x^,  Xo,  •••,  x„  in  the  ?;  equations  (1),  and  let  us  denote 
by  capital  letters  Ai,  •••,  Bi,  ■■■  the  cofactors  of  a^  ••-,  i^  •••  in  D. 

If  we  multiply  both  members  of  the  equations  (1)  in  order  by 
A^,  A.-,,  •••,  An,  and  add,  we  find 

(a^A^  +  a.-.A.  +  ■•■  +  a„An)xy  +  (b^A^  +  KA„  +  —  +  bnA„)x.;,  +  ■•■ 
+  (k.^^A^+  k„A.^  +  •••  +  k-nA„)x„  =  7\A^  +  r.-,A.,  +  •••  +  r„vl„. 

According  to  the  final  remark  of  Art.  232,  the  coefficient  of  xi  is  equal 
to  D,  and  the  coefficients  of  x„,  x.,,  •■■  x„  are  equal  to  zero.  The  right 
member  is  obviously  equal  to  an  nth  oi'd'er  determinant  which  may  be 
obtained  from  D  by  writing  i\,  r.,,  •••,  r„  in  place  of  a^,  a,,  •••,  «„•     Thus 


we  find 

(3)  Dx, 


^1     h 


Similarly  we  find 


(4) 


Dxo  = 


cin    r„ 


k, 


Dx„ 


«•>     h 


In  the  case  n  =  3  these  formulas  reduce  to  equations  (8)  of  Art.  209, 
if  we  put  x^  =  X,  X2  =  y,  Xg  =  z. 

From  (3)  and  (4)  we  obtain  the  values  of  the  unknowns  x,,  Xo,  •■■ ,  Xn 
if  we  divide  by  D,  provided  of  course  that  D  is  not  equal  to  zero. 


Art.  234]  HOMOGENEOUS   EQUATIONS  387 

Just  as  in  the  case  n  =  o  we  obtain  tlie  following  theorem: 

A  Si/stem  of  n  linear  equations  ivith  n  unknowns  has  a  single 
solution  composed  of  the  n  numbers  i'j,  .rg,  ■■•,  t„,  obtainable  fronn 
(3)  and  (4)  by  dividing  by  />,  provided  that  the  determinant 
D  of  the  coefficients  of  the  unknoivns  is  not  equal  to  zero. 

If  D  is  equal  to  zero  and  if  even  a  single  one  of  the  deter- 
minants occurring  in  the  right  members  of  (3^  and  (4)  is  dif- 
ferent from  zero^  the  system  (1)  lias  no  solution,  that  is,  the  n 
equations  (1)  are  not  consistent. 

The  following  supplementary  theorem  we  shall  state  with- 
out proof. 

If  D  =  0,  and  if  besides  all  of  the  nth  order  determinants 
which  are  written  in  the  right  members  of  (3)  and  (4)  are  also 
equal  to  zero,  then  system  (1)  has  infinitely  many  solutions. 
More  specifically  let  us  assume  that  among  the  minors  of  D 
there  exists  at  least  one  determinant  of  the  (n—  l)th  order  which 
is  not  equal  to  zero.  Then  a  single  one  of  the  n  unknowns  may 
be  left  arbitrary  and  all  of  the  others  may  be  expressed  in 
terms  of  that  one. 

234.  Homogeneous  equations.  If  rj  =  7*2  =  •••  =  r„  =  0, 
equations  (1)  of  Art.  233  assume  the  form 

a-^^x^  +  ^^3-2  +  ■  •  •  +  ^,.r„  =  0, 
n\  «2^i  +  V2  +  ••  •+  h^n=  0, 

«na^i  +  br^x^  +  •  •  •  +  k^x^  =  0. 

Such  a  system  is  said  to  be  homogeneous.  The  discussion  of 
Art.  210  may  be  generalized  to  the  case  of  n  unknowns 
without  any  trouble,  giving  rise  to  the  following  theorem. 

The  homogeneous  equations  (1)  have  no  solution  except  the 
obvious  one  x-^^  =  x^  =  ■••  =  Xn=  0,  if  their  determinant  D  is 
different  from  zero.  If  I>  =  0  these  equations  have  infinitely 
many  solutions  besides  the  obvious  one.  If  in  this  case  when 
D  =  0,  among  the  minors  of  D,  there  exists  at  least  one  deter- 
minant of  the  (jn  —  l)th  order  which  is  not  equal  to  zero,  then  the 


388       DETERMINANTS   OF    THE   n^^   ORDER      [Arts.  235-237 

ratios  of  the  n  unknowns^  x-^:  x^:  •■■  :  2;„,  ivill  he  determined 
unique!  1/  b>/  the  given  equations.  If  all  of  these  minors  vanish, 
even  these  ratios  luill  not  be  determined  uniquely. 

235.  Systems  of  linear  equations  with  more  equations  than 
unknowns.  Obviously  such  systems  will  ordinarily  not  be 
consistent.  We  might  solve  n  of  these  equations  for  the  n 
unknowns.  The  resulting  values  will  ordinarily  not  satisfy 
the  remaining  equations  which  were  not  used  in  the  solution. 
The  conditions  for  consistency  are  obtained  by  demanding 
that  the  values  of  the  unknowns  obtained  from  w  of  the 
equations  shall  also  satisfy  the  remaining  equations.  These 
conditions  can  always  be  expressed  most  conveniently  by 
ec^uating  to  zero  certain  systems  of  determinants. 

236.  Systems  of  linear  equations  with  fewer  equations  than 
unknowns.  Let  there  be  m  equations  and  m  -{■  n  unknowns. 
We  can  ordinarily  solve  these  equations  for  m  of  the  un- 
knowns in  terms  of  the  remaining  n  whose  values  remain 
perfectly  arbitrary.  Such  systems  therefore  usually  have 
infinitely  many  solutions. 

Thus  the  system 

(1)  x  +  y  -  z=\,    X-  y  +  -2z  =  2, 

may  be  solved  for  x  and  ij  in  terms  of  ~,  giving 

(2)  :,^__i(3  +  ~),   y  =  1(3  2-1). 

The  values  (2)  will  satisfy  (1)  no  matter  what  the  value  of  z  may  be. 

237.  Application  of  determinants  to  the  theory  of  elimina- 
tion.    A  quadratic  equation 

(1)  ax'^  +  bx  +  c  =  0 

has  two  roots  x^  and  x^.  Similarly  a  second  quadratic 
equation 

(2)  a'x"^  +  b'x-\-  c'  =  0 

also  has  two  roots  which  may  be  called  x^  and  x^.  Ordi- 
narily neither  of  the  two  roots  of  (2)  will  also  be  a  root  of 


Art.  237]  ELIMINATION  389 

(1),  but  in  special  cases  it  may  happen  that  this  is  the  case. 
What  condition  must  the  coefficients  of  the  two  quadratics 
satisfy  in  order  that  they  may  have  a  root  in  common  ? 

We  may  find  this  condition  as  folh)\vs.  If  tlie  equations 
(1)  and  (2)  have  a  root  in  common,  let  it  be  called  x.  Then 
X  will  satisfy  the  following  four  equations  : 

0  +  ax^  +  l>.r  +  c  =  0, 

.3.  0  +  a'x^  +  h'x  +  c'  =  0, 

ax^  +  bx^    +  C.V  +  (J  =  0, 

a'x^-hh'x^  +c's  +0  =0. 

But  we  may  regard  (3)  as  a  system  of  four  homogeneous 
equations  for  the  determination  of  the  four  (quantities 

Not  all  of  these  quantities  can  be  zero,  since  x^  is  equal  to 
unity,  even  if  x^,  x^^  and  x^  should  happen  to  be  equal  to  zero. 
Therefore  according  to  the  tlieorcm  given  in  Art.  234,  the 
determinant 


E  = 


must  he  equal  to  zero  if  the  equations  (1)  and  (2)  have  a  root 
in  common.  Our  argument  does  not  suffice  to  prove  the 
converse;  that  is,  if  E=0,  then  the  two  (quadratics  actually 
have  a  root  in  common.  Su(!h,  however,  is  actually  the  case, 
a  statement  which  we  shall  leave  without  proof.  U  is  called 
the  eliminant  or  resultant  of  the  two  quadratics  (1)  and  (2). 

EXERCISE    CVI 
1.    Solve  by  deteriniiiants 

:;./■  -f  -J//  +  1  ^  —  w  =  18, 
.')  ./■  +  1/  -  z  +'2  ir  =  0, 
2  .r  +  3  //  -  7 ;:  +  3  w  -  14, 
4  r  -  1  //  +  ;]--  .5  w  =  4. 


0 

a 

h 

c 

0 

a' 

h' 

a 

a 

h 

c 

0 

a' 

h' 

c' 

0 

390 


DETERMINANTS   OF   THE   n^^   ORDER       [Art.  237 


Discuss  consistency  or  inconsistency  of  tlie  following  systems  and  obtain 
the  most  general  solution  in  those  cases  in  which  the  systems  are  con- 
sistent. 


2. 


3. 


4. 


2x  +  7j  +  3z  =  1, 
4:x+2y-  z  =  -S, 

2x  +  y  +  3z  -1, 
4:x  +  2y  —  z  —  3, 
2x  +  y  — 42  =  4. 

x  +  y  +  -dz=:(), 
x  +  2y  +  2z  =  0, 
x  +  5y— z  —  0 


6. 


Qx  +  4:y  +  3z  -  Siio  =  0, 
x  +  2y  +  3z-^8  10  =  0, 
x-2  y  +  z  -12w  =  0, 
4:X  +  iy-z-2'i:U'  =  0. 

2x  -  y  +  3z  -2w  =  0, 
X  +  7  y  +  z  -  IV  =  0, 
3x  +  oy  —  5z  +  3  to  =  0, 
^x-3y  +  2z-w=^0. 


CHAPTER   Xril 

QUADRATIC    FUNCTIONS    OF    TV70    INDEPENDENT    VARI- 
ABLES   AND    SIMULTANEOUS    QUADRATIC    EQUATIONS 

238.  Integral  rational  functions  of  two  independent  vari- 
ables. A  function  of  the  two  independent  variables  x  and  y  is 
said  to  he  an  integral  rational  function,  if  it  can  be  expressed  as 
a  sum  of  a  finite  number  of  terms  of  the  form 

(1)  Ax'>/\ 

where  A  is  a  constant  coefficient  (that  is,  a  quantity  not  de- 
pending for  its  value  on  x  or  y^  and  ivhere  the  exponents,  r 
and  s,  are  positive  integers. 

Tlie  sum  of  the  exponents,  r  and  s,  is  called  the  degree, 
order,  or  dimension  of  tlie  terra  (1).  An  integral  rational 
function  is  said  to  be  of  degree,  order,  or  dimension  w  if  there 
occurs  in  it  at  least  one  term  of  order  n  and  no  term  of  degree 
greater  than  n. 

According  to  this  rnle  xr-,  xy,  and  y"^  are  terras  of  the  sec- 
ond order  in  x  and  y.  Some  students  find  it  difficult  to 
understand  why  the  terra  xy  should  be  classified  as  being 
of  the  second  order.  To  clear  up- this  difficulty,  consider  x 
and  y  as  the  numerical  measures  of  line-segments  (ex- 
pressed in  inches)  ;  then  each  of  the  three  terms  x"^,  xy,  and 
y"^  may  be  regarded  as  the  numerical  measure  of  an  area 
(expressed  in  square  inches). 

EXERCISE    evil 
State  the  degree  of  each  of  the  following  functions  of  x  and  y : 

1.  3x  +  4y-5.      4.    xy  +  X -\- y.     7.    xy'^.  10.    x'^y- -\- 3*. 

2.  x'^-y.  5.    x^  -  7/.  8.    xhj  +  y'^  -?,,x.      11.    x  -  xhj. 

391 


392     QUADRATIC  FUNCTIONS  OF  TWO  VARIABLES  [Art.  239 

Express  the  following  as  functions  of  two  independent  variables: 

13.  The  volume  and  surface  of  a  rectangular  parallelopiped  of  dimen- 
sions X,  X,  y. 

14.  The  volume  and  total  surface  of  a  right  circular  cylinder  of 
radius  r,  and  altitude  /(. 

15.  The  volume  and  total  surface  of  a  right  circular  cone  of  altitude 
11,  if  the  radius  of  its  circular  base  is  r. 

239.  Quadratic  function  of  x  and  y.  A?!  integral  rational 
function  of  the  seco7id  order  is  called  a  quadratic  function. 

Thus,  a  quadratic  function  can  contain  only  terms  involv- 
ing a:^,  xy.,  y^^  x,  ?/,  and  a  term  independent  of  x  and  y.  The 
most  general  quadratic  function  of  x  and  y  may  be  written 
as  follows 

(1)      fi^x,  y)  =Ax^  +  2  Hxy  +  By''  ^ -1  Gx  +  2  Fy  +  O, 

where  the  constant  coefficients  A,  B,  (7,  F,  (x,  H  may  have 
any  values  whatever.  We  have  used  the  notations  2  F., 
2  6r,  and  1  H  to  represent  the  coefficients  of  ^,  a;,  and  a;y,  in 
this  expression,  rather  than  F,  G-,  and  H.,  because  the  for- 
mulas assume  a  somewhat  simpler  aspect  when  we  use  this 
notation.     (Compare  the  corresponding  remark  in  Art.  71.) 

EXERCISE    CVIII 

In  each  of  the  following  examples,  determine  the  values  of  A,  B,  C, 
F,  G,  and  H. 

1.  f(x,  .'/)  =  B  3-2  +  4  xi/  +  7  //2  -  4  X  +  8  ?/  -  10. 

2.  /■(.!■.  //)  =  7  x^  -  ")  .r//  +  4  If-  +  -ix  -2y  +  G. 

3.  f(x,  y)  =  -  x-  -  2  x  +  \. 

4.  /(.r,  y)  =  X-  +  ?/2  —  4. 

240.  Composite    and    non-composite     quadratic    functions. 

Clearly,  the  product  of  two  linear  functions  of  x  and  y  is 
a  quadratic  function.  Therefore,  some  quadratic  functions 
can  be  expressed  as  products  of  two  linear  functions. 
Whenever  this  is  the  case  we  shall  say  that  the  quadratic 
function  is  composite  or  factorable.     We  shall  now  show  how 


Art.  •J40]       COMPOSITE   QUADRATIC   FUNXTIOXS 


393 


the   question   may  be    decided  whether  a    given   quadratic 
function, 

(1)  fix,  y)  =  Ax-^  +-2EXI/  +  Bf  +  2  Gx  + -2  Ft/ +  C, 

is  composite  or  not. 

Let    us    le-write    (1),    arrano'ing    the  terms  according  to 
descending  powers  of  ?/.      \Vc  fnul 

(2)  f(x,  y )  =  ^//2  +2(^Hx  +  F)y  +  Aj^  +  2  Gx  +  O. 
Let  us  assume  B=^0.     Then  we  may  write 


(3)      f(x,y}=B 


o  ,  .,Rx  +  F     ,  Ax^  +  2ax+  0 


>r  +  ^ 


]■ 


We  may,  for  a  moment,  think  only  of  the  way  in  which 
this  expression  depends  upon  y,  ignoring  the  fact  that  it  de- 
pends upon  X  also,  hi  order  to  do  this  it  will  help  us  to 
write 

where  we  have  put 


(•^) 


^Hx+F            A2'^+2ax+C 
P  =  '—B-^    ^  = ^ 


Tlie  quantity  in  the  square  bracket  of  (4)  may  be  resolved 
into  two  factors  of  the  first  degree  in  y  by  the  method  of 
Art.  68,  giving  first 


and  then 

(ti)    f(x,y)  =  B 


.E 


where,  on  account  of  (5), 

p''-Aq=  —MHx  +  Fy -  B(Ar^  +2ax+  (7)] 


^2 


or. 


(7)  pi-Aq=^[(E^-AB^x^  +  2(Fff-BG')x  +  F^-BC]. 
B^ 


394     QUADRATIC  FUNCTIONS  OF  TWO  VARIABLES  [Art.  240 

Since  the  factors  of /(a;,  ^)  as  exliibited  in  (6)  contain  the 
square  root  of  jo^  —  4  q,  these  factors,  although  rational  and 
of  the  first  degree  in  y,  will  in  general  be  irrational  func- 
tions of  X.  As  shown  by  (6)  and  (7),  the  only  irrationality 
which  these  factors  contain  is  the  square  root  of  the  integral 
quadratic  function 

(8)  Q {x)  =  (ir2  -  AB)x''  +  %FH-  B a)x  +  F'^  -  BO, 

and  the  square  root  of  Q{x}  will  be  a  rational  function  of  x 
if  and  only  if  the  two  linear  factors  of  Q  (a;)  are  the  same 
(see  Art.  152),  or,  as  we  may  express  it  briefly,  if  Q{x^  is  a 
perfect  square. 

We  may  rewrite  Q{x}  as  follows: 

(9)  Q(^x)  =  ax^-\-bx-{-c, 
if  we  put 

(10)  a  =  H^-AB,b=  2(Fff-  BG),  e=F^-  BO. 

The  two  linear  factors  of  Q{x^  will  be  the  same  if  and 
only  if 

(11)  b^-4:ac=0.        (See  Arts.  68  to  70.) 

If  we  introduce  the  values  (10)  of  a,  5,  c  into  this  condition 
(11),  we  find  , 

4(iFIf  -  Bay  -  4(H^  -  AB)(F^  -  BC}  =  0, 

or,  dividing  both  members  by  4  and  performing  the  indicated 
operations, 

F^IT^-2BFaH+  B'^G^ 

-  (i^2^2  _  ^^^2  _  ^CH^  +  AB^C)  =  0, 
which  reduces  to 

(12)  -  2  BFQH^  B^G-'^  +  ABF'^  +  BCm  -  AB^O^  0. 

Every  term  of  this  expression  contains  B  as  a  factor. 
Since  we  are  just  now  treating  the  case  B  ^0,  (12)  can  be 
satisfied  only  if 


Art.  1240]    DISCRIMINANT  OF  A  QUADRATIC  FUNCTION     395 

-2FaH+BG^  +  AF^+  CH^-ABC=0, 

or,  rearranging  the  terms  and  changing  all  of  the  signs,  if 

(13)  ABC+IFGR-AF^-BG^-  (77/2  =  0. 

Thus,  in  order  that  the  given  quadratic  function  may  be  a 
product  of  two  linear  functions,  its  coefficients,  A,  B,  C,  F, 
G,  £[,  must  satisfy  the  condition  (13),  at  least  if  B  is  different 
from  zero.  Moreover,  if  B^O,  and  if  A,  J5,  C,  F,  G,  H  do 
satisfy  this  condition  (13),  the  given  quadratic  function  can 
actually  be  expressed  as  a  product  of  two  linear  functions. 
The  factors  are  given  by  (6)  where,  if  (13)  is  satisfied,  the 
square  root  which  occurs  will  reduce  to  a  linear  function 
of  X. 

The  quantity  which  occurs  in  the  left  member  of  (13)  is 
usually  denoted  by  A  (pronounced  delta)  and  is  called  the 
discriminant  of  the  quadratic  function  (1)  of  x  and  y. 

Our  argument  only  covers  the  case  B  ^  0.  If  i?  =  0  we  have,  instead 
of  (1),  the  simpler  expression 

f(x,  y)  =  Ax^  +  2  Hxy  +  2  Gx  +  2  Fy  +  C. 

Let  us  assume  that  A  is  not  also  equal  to  zero  and  let  us  arrange  /(r,  y) 
according  to  descending  powers  of  x.  We  may  now  repeat  our  argu- 
ment essentially  as  above  merely  interchanging  the  part  played  by  the 
two  variables  x  and  y.  The  resulting  condition  for  factorability  will  turn 
out  to  be 

(14)  2FGn-  AF^  -  CH^^O. 

But  this  is  precisely  w-hat  (13)  would  reduce  to  in  the  case  5  =  0.  We 
conclude  that  the  condition  (lo)  may  be  used  even  it  B  =  0,  although 
our  original  method  of  finding  this  condition  was  not  applicable  to  this 
case. 

Our  argument  now  covers  all  cases  in  which  at  least  one  of  the  num- 
bers A  and  B  is  not  equal  to  zero.  It  remains  to  consider  the  case 
^  =  5  =  0.     The  function  (1)  reduces  to 

(15)  f(x,  y)=2  II xy  +  2  Gx  +  2  Fy  +  C, 

where  we  may  assume  /f  ^  0,  since  otherwise  the  function  would  reduce 
to  a  linear  function. 

If  (15)  is  a  product  of  two  linear  functions  only  one  of  the  factors 
can  contain  x.  For  otherwise  the  product  would  contain  a  term  involv- 
ing X-.     Similarly,  only  one  of  the  factors  can  contain  y.     Therefore,  if 


896     QUADRATIC  FUNCTIONS  OF  TWO  VARIABLES  [Aur.  240 

(15)  is  a  product  of  linear  factors  at  ail,  it  must  be  equal  to  a  product  of 
the  form 

(16)  {px -\- q){r7j -{-  s). 

Tims,  in  this  case,  (15)  must  l)e  equal  to  (16),  that  is,  we  must  have 
•  2  Ilxy  +  2  Gx  +  2  Fi/  +  C  =  prxy  +  psx  +  qry  +  qs 

for  all  values  of  x  and  y.     But  this  is  possible  only  if 

(17)  2II^pr,   2G^ps,   2  F  =  qr,    C  =  qs. 

Since,  in  the  case  under  consideration,  we  also  have  A  =  B  =  0,  the  left 
member  of  (1:5)  will  assume  the  value 

'qr\( P><\f  pf\  p-r'^  _1     „    .,         1 


'(f^jlf  )(?■)  ~  9^^  =  |PV'«  -  ^P'V 


0. 


Consequently  condition  (13)  is  satisfied  by  the  coefficients  of  a  composite 
quadratic  function  even  ii  A  =  P>  —  0.  Conversely,  if  the  coefficients  of  a 
quadratic  function  satisfy  the  conditions  (13)  and  A  =  B  =  0,  we  have 

(18)  ^  =  jB  =  0,   2  FGH  -  CH^  =  0,   H  =^0, 

and  we  can  actually  find  four  numbers  p,  q,  r,  and  s  which  satisfy  equa- 
tions (17).     It  suffices  for  this  purpose  to  put 

p  =  2II,    q  =  2F,    r^l,   s  =  ^  =  -^ 
f  '    /  H      2F 

where  the  two  values  obtained  for  s  are  equal  on  account  of  (18),  and 
where  F  is  assumed  to  be  different  from  zero.  If  F  should  be  equal  to 
zero,  we  should,  according  to  (18),  find  C  =  0  also,  since  H  ^  0,  and  (15) 
would  reduce  to 

f(x,  y)  =  2  Ilxy  +  2  Gx  =  2  x  (Hy  +  G^ 

the  composite  nature  of  which  is  evident  by  insjiection. 

We  liave  proved  the  following  important  theorem: 
An  integral  rational  quadratic  function 

( 1 9)  Ax"^  +  2  Hxy  +  Bf  +  2  G.v  +  2  Ft/ +  0 

may  he  expressed  as  a  product  of  two  integral  rational  functions 
of  the  first  degree.,  if  and  only  if  the  coefficients  satisfy  the  con- 
dition that  the  discriminant 

(20)  A  ^  ABC +2  Faff-  AF'^  -BG"^-  Gff'^ 

is  equal  to  zero. 

It  is  easy  to  apply  this  criterion  to  any  given  quadratic 
function.     It  is  also  easy  to  actually  find  the  factors  if  the 


Art.  240]       COMPOSITE   QUADRATIC    FUNCTIONS  397 

condition  is  satisfied.     It  is  sufficient    for  tliis   purpose  to 
apply  the  general  methods  of  this  article. 

If  tlie  function  (10)  is  homogeneous,  that  is,  if  it  contains 
no  terms  of  degree  lower  than  the  second,  we  have  (7=0, 
F=  0,  (r  =  0,  and  therefore,  according  to  (20),  A  will  also 
be  equal  to  zero.  Therefore,  every  homof/eneous  quadratic 
function  may  be  resolved  into  integral  rational  factors  of  (he 
first  degree. 

The  factors  of  (19)  need  not  be  real.  Thus  the  factors  of  the  homo- 
geneous function  x-  +  y'^  are  .r  +  ///  and  :r  —  i;/. 

EXERCISE   CIX 

1.  Is  2 x^  —  xy  —  3 _?/■-  +9 X- +  4^  +  7  a  composite  function ?  If  so, 
find  its  factors. 

Solution.  We  have  .4=2,  73  =  -  3,  C  =  7,  F  =  2,  G  =  f,  H  =  -  j, 
and,  therefore,  find  A  =  0.  Therefore  the  function  is  composite.  To  find 
the  factors  follow  the  general  method  of  Art.  240.  The  given  function 
is  equal  to 

^      .r/^  l-'--^)-'      (•f-4)-^  +  12(2x'^  +  9x  +  7)-| 

__  .^  r/2-  +  0  y  -  4\^      25  x^  +  100  X  +  lOOl 
~      '  Lv  6  /  36  J 

=  -  u(^  +  Qi/-  -^r  -  (5  X  + 10)2] 

=  -  1^2  [x  +  6  .y  -  4  +  5  X  +  10]  [x  4-  6  J/  -  4  -  (.5  x  +  10)] 

=  -J,(Gx+()^  +  G)(-4x  +  Gy-14)=-(.t-f//+l)(-2x  +  3^-7) 

=  (x  +  i/  +  l)(2x-3y  +  7). 

Thus,       2 a;2  -  x/y  -  3  y-  +  9  X  +  4  y  +  7  =  (x  +  y  +  1) (2  X  -  3 3/  +  7). 

The  result  may  be  checked  by  multiplying  out  the  product  in  the 
right  member. 

In  each  of  the  following  examples  state  whether  the  given  function 
is  composite  or  not.     Find  the  factors  if  they  exist. 

2.  -  x^ +  4:Xij  -  ij^ -4:^/2  X +  2^2 y -11. 

3.  y-  —  xy  —  Q  x^  +  y  —  'i  X. 

4.  4  x^  —  4  X2/  +  ^2  +  4  X  —  2  y. 


898     QUADRATIC  FUNCTIONS  OF  TWO  VARIABLES  [Art.  241 

241.  The  values  of  a  quadratic  function.  In  order  to 
obtain  some  idea  of  tlie  values  wliich  a  quadratic  function 
takes  on  for  various  values  of  x  and  ?/,  we  formulate  tlie 
question  :  what  are  all  of  the  values  of  x  and  y  which  cause 
the  function  to  assume  the  given  value  h  ?  That  is,  what 
are  the  values  of  x  and  y  for  which 

(1)  ^z2  +  2  Hxy  +  By'^+2ax^2Fy+0=V'^ 

These  values  are  obviously  the  same  as  those  which  cause 
the  new  quadratic  function 

(2)  Ax'^  +  IHxy  +  By'^^-^ax+IFy+O-h 

to  assume  the  particular  value  zero.  Since  (2)  is  a  function 
of  the  same  kind  as  the  left  member  of  (1),  our  question 
may  be  regarded  as  answered  if  we  can  only  solve  the  fol- 
lowing, somewhat  simpler,  problem.  To  find  all  of  those 
pairs  of  values  for  x  and  y  which  will  satisfy  any  given  equa- 
tion of  the  form 

(3)  Ax"^  +  IHxy  +  By^ -^  2  ax  +  ^Fy  +  C  =  0. 

Every  pair  of  numbers  which,  when  substituted  for  x  and 
y,  will  satisfy  equation  (3)  is  called  a  solution  of  the  equa- 
tion.    The  equation  is  called  a  quadratic  equation. 


EXERCISE  CX 

In  each  of  the  following  examples  test  the  given  pair  of  numbers,  to 
see  whether  or  not  they  constitute  a  solution  of  the  given  equation. 
Give  the  details  of  your  work. 

1.  X-  —  4  z^  =  0 ;  X  =  2,  ?/  =  1. 

2.  x2  +  2  x?/  +  ?/2  =  5 ;  X  =  1,  2/  =  1. 

3.  x^  —  ?/2  _  0 ;  X  =  any  number,  y  =  x. 

4.  x2  +  »/2  :^  0 ;  X  =  1,  ?/  =  «■  =  V"^r. 

5.  x2  —  xij  +  ?/2  =  7 ;  X  =  1,  ?/  =  0. 

6.  (X  -  ?/)2  +  2x  -  3/  =  2;  X  =  0,  2/  =  2. 


Art.  242]  EXISTENCE   OF   SOLUTIONS  399 

242.  The  existence  of  solutions  of  a  quadratic  equation. 
Every  quadratic  equation 

(1)  A.i-  +  2  Hxy  +  %2  +  ^ax+  IFij  +  6'=  0 

Aas  infinitely  many  solutions.  For  we  may  substitute  for  x 
any  number  we  please  (real  or  complex),  and  then  obtain, 
if  B  =^  0,  two  corresponding  values  for  y  by  solving  the 
equation  (1),  which  then  becomes  a  quadratic  equation  with 
a  single  unknown,  for  y.  Thus  (1)  may  be  tliought  of  as 
defining  y  as  a  two-valued  function  of  x,  provided  that  B  is 
not  equal  to  zero.  If  B  =  0  and  if  at  least  one  of  the  coef- 
ticients  ^and  F  is  not  equal  to  zero,  we  find  that  (1)  defines 
3/  as  a  one-valued  function  of  x.  Finally  if  B,  F,  and  II hre 
all  equal  to  zero,  the  equation  (1)  does  not  contain  any  y  ; 
it  will  be  satisfied  by  the  two  values  of  x  which  are  roots  of 
the  equation  Ax^ -\-  2  G-x-\-  C=  0;  with  each  of  these  values 
of  X  may  be  associated  arbitrary  values  of  y. 

Although,  as  we  have  just  seen,  every  equation  of  the 
form  (1)  has  infinitely  many  solutions,  it  does  not  follow 
that  these  solutions  are  real.  There  actually  exist  equations 
of  this  sort  which  have  no  real  solutions  at  all. 

Thus  the  equation  x-  +  //-  +  4  =  0  has  no  real  solutions.  For,  if  x 
and  y  are  real  numbers,  x^  and  y'^  cannot  be  negative,  and  therefore 
a;2  +  ^-  +  4  cannot  assume  any  value  less  than  4  for  real  A'alues  of  x  and 
y.  The  equation  x^  +  ^^  _  o  ^as  one  and  only  one  real  solution,  namely 
X  =  ^  =  0. 

We  have  shown  in  Art.  240  how  the  left  member  of  (1) 
may  be  resolved  into  two  factors  of  the  first  degree  in  y  if 
B  ^  0.  In  order  that  the  equation  (1)  ma}^  be  satisfied,  one 
of  these  factors  at  least  must  vanish.  Therefore,  if  we  use 
the  notations  of  Art.  240,  we  find  the  following  explicit 
expression  for  ?/  as  a  function  of  x  : 


(2)  ^^-p±y-4y^ 

where 

(.3)     p  =  hnx+F~),  q='^(iAx^  +  2ax+C\  B=^0. 
B  B 


400     QUADRATIC  FUNCTIONS  OF  TWO  A^ARIABLES  [Art.  242 

These  formulas  show  clearly  the  two- valued  character  of 
this  function,  since  we  may  use  either  the  plus  or  minus  sign 
in  (2).  We  may  also,  and  in  many  respects  this  is  desirable, 
regard  (2)  as  defining  two  one-valued  functions  rather  than 
a  single  two-valued  function.  The  resulting  tivo  functions 
are  rational  or  irrational  according  as  p^  —  4q  is  a  perfect 
square  or  not,  that  is,  according  as  the  discriminarit  A  o/"  (1) 
is  or  is  not  equal  to  zero. 

The  case  B  =Q  is  easily  settled.  We  shall  leave  the  dis- 
cussion of  this  case  as  an  exercise  for  the  student. 

The  formulas  (2)  and  (3)  give  us  the  general  solution  of 
equation  (1)  in  the  case  B  ^  0.  If  we  substitute  in  them 
for  X  any  number  whatever,  and  then  compute  the  corre- 
sponding values  of  ?/,  all  pairs  of  numbers,  x  and  i/,  obtained 
in  this  way  are  solutions  of  (1). 

EXERCISE  CXI 

Solve  each  of  the  following  equations  for  ?/  as  a  function  of  x  and  dis- 
cuss the  following  questions.  Is  the  resulting  function  one-valued  or 
two-valued  ?  If  it  is  two-valued  in  general,  are  there  any  particular 
values  of  x  for  which  the  two  values  of  y  coincide?  For  what  real  values 
of  X  will  the  resulting  values  of  y  also  be  real? 

1.  X-  +  _y2  =  4.  8.    X-  +  if-  =  a^. 

2.  X-  -  y-  =  4. 

3.  xy  —  5. 

4.  4x-  -\-  y'^  =  10. 

5.  a,-2  +  4  7/2=  16.  ^^-    ~ 

^  11. 

7.    x^  -  4y-  =  16. 

18.  (// -  ^•)'-  =  4;>(.r- A). 

19.  {x  -  hy^  =  ip(y  -  k). 

20.  (x  -  'Ay  +  (y  -   iy  =  5":  ^^     (x  - /,y  _  (y  -  /.)-  ^  ^ 

21.  (x  -  hy  +  (y  -  l-y  =  «2,  •   "      a-2  b- 

24.  _I:^^0_%I.'/-/-V^  =  l. 

(r  h- 

25.  -  x2  -H  4  xy  -  ?/2  -  4  \/2  x  +  2  V2  ^  -  11  =  0. 

26.  y^  -  x^  -  6  X-  4-  y  -  3  X  =  0. 


Akt.  21:3]        GRAPH   OF   QUADRATIC    EQUATION  401 

243.  Graph  of  a  function  defined  by  a  quadratic  equation  in 
X  and  y.  Whenever  a  function,  detinecl  by  means  of  a  quad- 
ratic equation  in  x  and  y,  is  real,  that  is,  if  tlie  quadratic 
equation  has  real  solutions,  these  may  be  plotted  as  points 
in  accordance  witli  the  metliod  which  we  have  used  so  fre- 
quently. The  general  question  as  to  the  nature  of  the 
graphs  obtained  in  tins  way,  while  not  very  dilTicult,  is 
usually  reserved  for  the  course  in  analytic  geometry. 
There  are,  however,  several  special  cases  in  which  it  is  quite 
easy  to  draw  the  graphs.  We  shall  now  discuss  some  of 
these  cases,  a  few  of  which  have  appeared  already  in  this 
book  in  a  different  connection. 

Case  1.  The  graph  of  a  quadratic  equation  in  x  and  y  con- 
sists of  a  pair  of  straight  lines  if  the  discriminant  A  is  equal  to 

zero. 

For  we  have  seen  in  Art.  240  that,  in  this  case,  the  quad- 
ratic function  is  a  product  of  two  linear  functions,  say 
a^T  +  b^g  +  c^  and  a^x+b^g  +  c^  ;  and  we  have  also  shown  how 
these  linear  factors  may  be  found.  The  values  of  x  and  y, 
which  cause  the  quadratic  function  to  vanish,  must  make 
at  least  one  of  the  factors  equal  to  zero.  But  the  locus  of 
all  of  those  points,  whose  coordinates  cause  a  linear  function 
of  X  and  g  to  vanish,  is  a  straight  line.  Therefore,  the 
graph  of  a  quadratic  equation  with  a  vanishing  discriminant 
consists  of  the  two  straight  lines 

a^x  +  b^g  -(-  (?!  =  0,  a^r  -f-  b.^r/  +  c.^  =  0, 

whose  equations  are  obtained  by  equating  to  zero  the  linear 
factors  of  the  quadratic  function. 

Case  2.  If  B  =  0,  g  becomes  a  rational  function  of  x. 
If  A  =  0,  X  becomes  a  rational  function  of  g.  \\\  either  case, 
the  graph  may  be  constructed  by  the  methods  of  Art.  140, 
special  attention  being  devoted  to  the  poles  of  these  rational 
functions  whenever  they  have  any  poles.     (See  Art.  139.) 

For  illustrations  see  Art.  140. 


402     QUADRATIC  FUNCTIOXS  OF  TWO  VARIABLES  [Art.  243 

Case  3.  If  F  =  G-  =  H  =  0,  the  locus  of  the  equation  is 
symmetric  with  respect  to  both  the  x-axis  and  y-axis.  In  this 
case  the  origin  is  called  the  center  of  the  locus. 

Proof.     The  equation,  in  this  case,  reduces  to 
(1)  Ax'^  +  %2  ^  c=0. 

If  this  equation  is  satisfied  by  a  pair  of  numbers  (a;,  j/),  it 
will  also  be  satisfied  by  the  pair  (a;,  —  y'),  since  only  the 
second  power  of  y  occurs  in  the  equation  and  since 
(~~  I/)^  =  (+  ^)^-  ^^t  ^li6  ^^^o  points  (2;,  ^)  and  (.t;,  —  y} 
are  symmetrically  situated  with  respect  to  the  a:-axis.  A 
similar  argument  shows  that  for  any  point  (x,  y^  which  is 
on  the  graph  of  (1),  there  exists  a  second  point  (— a;,["?/) 
which  is  also  on  the  graph  and  which  is  symmetric  to  the 
first  point  with  respect  to  the  y-axis. 

It  is  easy  to  see  what  the  graph  of  (1)  will  look  like  in 
the  various  cases.  If  B  =  0,  (1)  does  not  contain  any  y 
term  at  all,  and  solution  of  the  equation  for  x  shows  that  the 
graph  consists  of  a  pair  of  lines  parallel  to  the  y-axis.  These 
lines  may  be  real  and  distinct,  coincident,  or  imaginary. 
Of  course  in  this  case  A  is  equal  to  zero. 

li  B  =^  0,  we  may  rewrite  (1)  as  follows 


y^  = 

Ax^-\-  0         A  ^ 
B       =      B^- 

«/2  =  mx'^  +  w, 

0 
'  B' 

or 

(2) 

if  we  use  the  abbreviations,  m  and  n,  for  —  A/B  and  —  C/B 
respectively.  The  appearance  of  the  graph  depends  essen- 
tially on  the  values  of  m  and  w,  and  more  especially  upon 
the  signs  of  these  numbers. 

Case  3  a.  If  m  =  0,  the  graph  of  (2)  reduces  to  a  pair 
of  straight  lines,  parallel  to  the  a;-axis,  namely  the  lines  for 
which 

y  —  ±  Vw. 


Art.  243]  GRAPH   OF  QUADRATIC   EQUATION  403 

These  lines  are  real  and  distinct,  coincident,  or  imaginary, 
according  as  n  is  positive,  zero,  or  negative. 

Case  ?>  h.  \i  )i  =  0,  the  graph  of  (2)  reduces  to  a  pair  of 
straight  lines  throngli  the  origin,  namely  y  =  ±  Vma\  these 
lines  being  real  and  distinct  only  if  ni  is  positive. 

Case  3  c.  If  m  and  n  are  both  negative,  the  right  mem- 
ber of  (2)  will  be  negative  for  all  real  values  of  x.  Conse- 
quently, in  this  case,  no  real  values  of  i/  will  correspond  to 
real  values  of  x;  the  equation  has  no  real  solutions  and  it  has 
no  real  graph. 

There  remain  three  cases,  namely  Case  3  d  when  w  <  0, 
%  >  0,  Case  3  e  when  m  >  0,  w  <  0,  and  Case  3/  when  m  >  0, 
n>0. 

Case  Z  d.     w  <  0,  w  >  0.     We  may  write,  in  this  case, 

(3)  m  =  —  k\  n  =  k^a^, 

where  a  and  k  may  be  regarded  as  positive  numbers  whose 
values  may  be  computed  from  (3)  in  any  case  where  m  and 
n  are  given,  m  as  a  negative  and  w  as  a  positive  number. 
Then  (2)  becomes 

y  =  —  k^x"^  +  k'^aP-  =  k'^(^a^  —  a;^), 
whence 


(4)  y  =  ±  kVa^  —  x\ 

If  x^  >  a^  a^  —  x^  will  be  negative  and  therefore  y  will 
be  imaginary,  as  shown  by  (4).  Therefore,  there  are  no 
points  on  the  graph  whose  distances  from 
the  ?/-axis  are  greater  than  a.  Let  us 
mark  the  points  A  and  A'  on  the  a;-axis 
(see  Fig.  70)  whose  abscissas  are  equal 
to  +  a  and  —  a  respectively.  If  we  draw 
lines  parallel  to  the  y-axis  through  these 
points,  we  know  then  that  no  point  of 
the  graph  can  be  outside  of  the  strip  inclosed  by  these  two 
lines.  The  points  A  and  A'  themselves  are  points  of  the 
graph.     For  if  we  put  a;  =  ±  a  in  (4),  we  find  y  =  0. 


404     QUADRATIC  FUNCTIONS  OF  TWO  VARIABLES  [Art.  243 


For  X  =  0,  equation  (4)  gives  y  =  ±  ka.  Let  us  construct 
the  points  B  and  B'  on  the  y-axis  (Fig.  70)  whose  ordi- 
nates  are  equal  to  +  ka  and  —  ka  respectively.  Both  of 
these  points  are  on  the  graph  of  equation  (4).  For  any 
value  of  X  which  lies  between  —  a  and  +  a,  we  find  from 
(4)  two  values  of  y,  of  the  same  absolute  value  but  of  oppo- 
site sigh,  thus  proving  again  that  the  graph  is  sjanmetric 
with  respect  to  the  a^-axis.  Moreover  as  x  increases  from  0 
to  a,  the  numerical  value  of  y  decreases  from  ka  to  0. 

If  ^  <  1,  the  case  assumed  in  constructing  Fig.  70,  OB  is 
less  than  OA.  If  ^  >  1,  we  should  find  OB  >  OA.  If  ^  =  1, 
we  have  OB  =  OA^  and  it  looks  as  though  the  curve  would 
be  a  circle  with  0  as  its  center.  This  is  actually  the  case. 
For  if  ^  =  1,  (4)  becomes 


whence 


(5) 


?/  =  ±Vt 


y^  =  a^  —  3? 


.r2 


or 


Fig.  71 


rjfi    J^    y^   —  (jfi^ 

But,  if  X  and  y  are  the  coordinates  of  a  point 
P  (see  Fig.  71),  x^  -\-  y'^  will  be  the  square  of 
the  distance  from  the  origin  0  to  the  point 
P.     If  the  point  P  moves  in  such  a  way  that 

x^^f-  =  OP 


always  remains  equal  to  the  same  number  a^^  it  must  remain 
on  the  circumference  of  a  circle  whose  center  is  0  and 
whose  radius  is  equal  to  a.  Therefore,  the  graph  of  (5)  is 
indeed  a  circle  whose  center  is  at  the  origin  and  whose  radius 
is  equal  to  a. 

When  k  is  not  equal  to  unity,  the  graph  of  (4)  is  called 
ill!  ellipse.  The  line-segments  AA'  and  BB'  are  called  the 
principal  axes  of  the  ellipse,  the  larger  one  being  the  major 
axis  and  the  smaller  the  minor  axis. 

Case  3  e.     m  >  0,  w  <  0.     In  this  case  we  put 
m  =  k^,  n  =  —  k'^a^, 


Art.  243]        GRAPH    OV   QUADRATIC   EQUATION 


406 


so  that  (2)  becomes 


whence 


?/2  =  Hi:2  _  ^2^2  _  ^2^  2;2  —  a^), 


(6) 


y  =  ±  k^aP- 


i'lG.  7'^ 


The  discussion  of  this  case  is  ^y 

quite  similar  to  that  of  Case  3  d. 
But  this  time  the  values  of  x 
which  give  rise  to  imaginary 
values  of  y  are  the  small  values, 
namely  those  for  which  a?  <  a^. 
The  vertical  strip  of  width  2  a, 
between  A  and  A\  instead  of  in- 
cluding all  of  the  points  of  the 
curve  as  it  does  in  Case  3  d,  does  not  include  any  point  of 
the  curve.  The  curve  has  two  infinite  diverging  branches, 
as  illustrated  in  Fig.  72,  and  is  called  a  hyperbola.  Case  3/ 
also  leads  to  a  hyperbola,  but  located  in  a  different  way.  It 
may  be  obtained  by  rotating  Fig.  72  through  an  angle  of  90°. 

Case  4.  If  H=  0,  if  A  and  B  are  both  different  from  zero, 
while  at  least  one  of  the  numbers  F  and  (7  is  not  equal  to  zero, 
the  (jraph  will  have  the  same  general  characteristics  as  in  Case 
3,  but  the  tivo  axes  of  symmetry  will  not  both  coincide  with  the 
coordinate  axes.  They  will  merely  he  parallel  to  the  coordinate 
axes. 

The  following  illustrative  example  will  show  how  the 
axes  of  symmetry  may  be  obtained,  by  the  process  of  com- 
pleting the  squares,  and  how  the  locus  of  such  an  equation 
may  be  plottc^l. 

Example.     Discuss  4  a;^  +  sy^  _  g  x  —  4  y  +  -4  —  0. 

Solution.  We  collect  the  terms  involving  x  and  complete  tbe  square, 
and  proceed  similarly  for  the  terms  involving  //.     This  gives 

(7)  4  x^  -  8  .r  +  4  +  if'  _  ]  ,/  +  4  =  4  +  4  -  4  =  4. 

(8)  4(x  -  1)^  +  0/  -  -ly  =  4. 


406     QUADRATIC  FUNCTIONS  OF  TWO  VARIABLES  [Art.  243 


+y 

+!/' 

B 

r 

\ 

P 

1 

A 

,"' 

/ 

A 

M' 

V 

y 

0 

B' 

il 

I 

Fig.  73 


and  therefore 


Let  us  mark  the  point  O'  whose  co- 
ordinates are  x  =  1,  y  =  2  (see  Fig. 
73)  and  ciroose  a  new  coordinate  sys- 
tem with  O'  as  origin,  the  new  (x' 
and  y')  axes  being  parallel  to  the 
old  (x  and  y)  axes.  Let  P  be  a  point 
whose  coordinates  referred  to  the  old 
coordinate  system  are  x  and  y,  and 
let  its  new  coordinates  be  called  x'  and 
y'.     Then  we  have  (see  Fig.  73) 

X  =  OM,         y=  MP, 
x'  =  O'M',      y'  =  jWP, 
OB'  =  1,        B'O'  =  2, 


so  that 
(8) 


x=  0M==  OB'  +  B'M  =  0B'+  O'M'  =  1  +  ar', 
y  =  MP  =  MM'  +  M'P  =  B'0'  +  AFP  =2  +  y', 


x-  1 


^,  y 


-'  =  ?/• 

Let  us  substitute  these  values  in  (8).     AVe  find 
4  x'-  +  ?/'2  =  4 

which  is  an  equation  of  the  form  discussed  under  Case  3.  Thus  the  x' 
axis  and  y'  axis  are  the  axes  of  symmetry  of  this  curve,  and  the  point  0' 
is  its  center.  We  may  continue  the  discussion  as  in  Case  3  and  verify 
that  the  loans  is  the  ellipse  ABA'B'  shown  in  Fig.  73. 

Case  5.     If  F  =  G-  =  Q^  that  is,  if  the  equation  contains  no 
terms  of  the  first  degree,  the  origin  is  the  center  of  the  locus. 

Such  an  equation  lias  the  form 
(10)  Ax"^  +  2  m-g  +  Bg^  +(7=0. 

If  X  and  g  constitute  a  pair  of  numbers  which  satisfy  this 
equation,  the  numbers  (—.?-,  —  ^)  will  also  form  such  a  pair. 
But  the  origin  is  always  halfway  between  two  points  whose 
coordinates  are  (x,  g^  and  (  —  a;,  —  ?/)  .  Consequently  the 
locus  of  any  equation  of  the  form  (10)  has  the  property 
that,  to  every  point  P  which  is  on  the  locus,  there  corre- 
sponds another  point  P'  also  on  the  locus,  sucli  that  the 
segment  PP'  is  bisected  at  the  origin.  This  is  what  we 
mean  by  saying  that  the  origin  is  the  center  of  the  curve. 


Art.  244]     LINEAR  AND  QUADRATIC   EQUATIONS  407 

Case  6.  This  is  the  general  case  which  we  shall  not  dis- 
cuss in  detail.  It  may  be  stated,  however,  without  proof, 
that  the  graph  will  always  be  an  ellipse,  a  parabola,  a  hyper- 
bola, or  a  pair  of  straight  lines.  The  distinction  between 
this  case  and  the  others  which  we  have  discussed  lies  merely 
in  the  fact  that  the  axes  of  symmetry  of  the  curve  will  not 
be  related  to  the  .r-axis  and  y-axis  in  such  a  simple  manner. 

All  of  the  curves,  which  may  be  obtained  as  loci  of  quad- 
ratic equations  in  x  and  y,  are  known  by  the  collective  name 
conies. 

EXERCISE    CXIl 
Plot  the  loci  of  the  equations  given  in  Exercise  CXI. 

244.  Solution  of  a  system  of  simultaneous  equations  one  of 
which  is  linear  and  one  of  which  is  quadratic.  All  of  the 
points  whose  coordinates  satisfy  an  equation  of  the  first 
degree 

(1)  Ir  +  7ni/  +  71  =  0 

are  on  a  certain  straight  line.  All  of  those  whose  coordi- 
nates satisfy  an  equation  of  the  second  degree 

(2)  Ax^  +  2  my  +Bif  +  2  a.r  +  '2Fij+C=0 

are  on  the  conic  (parabola,  ellipse,  hyperbola,  or  a  pair  of 
straight  lines)  which  is  the  locus  of  (2).  The  points  whose 
coordinates  satisfy  both  equations,  if  there  are  any,  will  be 
the  common  points,  or  points  of  intersection,  of  the  straight 
line  and  the  conic.  Therefore  we  may  find  the  real  solutions 
which  the  two  equations  (1)  and  (2)  have  in  common,  hy  draw- 
ing the  line  and  the  conic  represented  hy  these  two  equations 
individually,  and  then  measuring  the  coordinates  of  their  points 
of  intersection. 

The  common  solutions  of  the  two  equations  may  also  be 
found  quite  easily  by  algebra.  We  solve  (1)  for  y,  and  sub- 
stitute the  resulting  value  of  y  in  (2),  thus  obtaining  a  quad- 
ratic equation  for  x  alone.     If  Xj^  and  x^  are  the  two  roots  of  this 


408     QUADRATIC  FUNCTIONS  OF  TWO  VARIABLES  [Art.  244 

quadratic  equation,  there  will  correspond,  by  means  of  (1), 
to  each  of  these  values  of  a;  a  value  of  y.  If  y^  and  y^  are 
these  values  of  ?/,  the  common  solutions  of  (1)  and  (2)  will 
be  (Xy,  y-[)  and  (x^,  y,^.  This  process  would  fail  only  if  m 
were  equal  to  zero.  In  that  case  we  solve  (1)  for  x^  and 
substitute  the  resulting  value  in  (2),  obtaining  a  quadratic 
for  y  alone. 

We  always  obtain  two  solutions  (a^j,  y-^  and  (x^-,  y^)  which 
may,  however,  as  in  the  simple  case  of  Art.  70,  be  real  and 
distinct,  coincident,'  or  imaginary.  The  corresponding  loci 
actually  intersect  in  distinct  points  only  when  the  solutions 
are  real  and  distinct.  If  the  two  solutions  coincide,  the 
straight  line  is  tangent  to  the  conic.  If  the  solutions  are 
imaginary,  the  straight  line  and  the  conic  do  not  intersect  at 
all. 

EXERCISE    CXlll 

Solve  the  following  systems  of  simultaneous  equations,  and  verify 
your  results  geometrically  by  graphs : 

'   \xrj  =  187.  '  [xy  =  45. 

^     |x-//  =  G,  g  |a;2  +  ?/2-8.r-4f/-5  =  0, 

[  xy  =  91.  '  [  3  X  +  4  ?/  =  5. 

^     |.r+  y=  11,  ^  ^.^  +  j/  =  4, 


I     X-  +  ^2  =  73.  [  X2  +    7/2  _   1(3. 

|.i:-7/=4,  \x  +  y  =  a, 

■    I  X'  +  y'=  208.  [  x'2  +  y/2  =  «2. 

9.    What  must  be  the  value  of  h  in  terms  of  m  and  r,  in  order  that 
the  two  solutions  of  the  system 

y  =  mx  +  b,   a;2  +  //2  =  r2 
Tuay  be  identical? 

Hint.     The  quadratic  equation  for  x,  obtained  by  eliminating  y,  must 
have  equal  roots.     (See  Art.  70  and  Exercise  XXIV.) 

10.    What  must  be  tiie  value  of  r  in  terms  of  a,  b,  and  m,  in  order  that 
the  two  solutions  of  the  system 


may  coincide? 


3:-  ,  y-      1 
y  =  mx  +  c,    -  +  ^  =  1 


Arts.  245,  246]      SIMULTANEOUS   QUADRATICS  409 

245.  Simultaneous  quadratics.  The  problem  of  finding 
those  pairs  of  numbers,  x  and  //,  which  satisfy  eacli  of  two 
quadratic  equations 

(1)  Ax^^1Hxij^Bif^2ax-\-1Fy-^C=^, 

(2)  Ax^  +  2  E'xy  +  B' if  +'2a'x  +  2  F y  +  C'  =  0 

is  equivalent  graphically  to  that  of  finding  the  coordinates 
of  those  points  in  which  the  two  conies  intersect  which  are 
obtained  by  making  the  graphs  of  (1)  and  (2).  Conse- 
quently we  have  a  graphic  solution  of  the  problem  immedi- 
ately available.  We  need  merely  draw  the  graphs  of  the 
two  equations,  determine  their  points  of  intersections,  and 
measure  the  coordinates  of  these  points.  However,  this 
graphical  method  will  only  furnish  the  real  solutions  of  the 
problem,  and  these  only  approximately. 

We  might  think  of  attacking  the  algebraic  sohition  of  the  problem 
by  means  of  the  following  direct  but  rather  clumsy  method.     Arrange 

(1)  according  to  descending  powers  of  ij,  and  solve  the  resulting  equa- 
tion, which  is  a  quadratic  in  y,  for  y  as  a  function  of  x.  The  resulting 
expression  for  y  is  given  by  ("i)  and  (3)  of  Art.  242.  If  we  substitute 
this  value  of  y  in  equation  (2),  we  obtain  in  general  an  irrational 
equation  for  x.  If  this  be  rationalized  (see  Art.  154),  the  resulting 
equation  for  x  turns  out  to  be,  in  general,  an  equation  of  the  fourth  de- 
gree. Since  such  an  equation  has  four  roots  (Art.  126),  the  solution  of 
this  equation  will  furnish  four  values  of  x. 

In  similar  fashion,  elimination  of  x  between   the  equatinns  (1)  and 

(2)  gives  rise  to  an  equation  of  the  fourth  degree  for  y,  the  oolution  of 
which  furnishes  four  values  of  y. 

The  four  values  of  x  and  the  four  values  of  //,  obtained  in  this  way, 
can  be  combined  into  jiairs  in  sixteen  ways.  But  not  all  of  these  pairs 
are  solutions  of  (1)  and  (2).  It  remains,  therefore,  to  ascertain,  by 
actual  substitution  in  (1)  and  (2),  which  of  these  sixteen  combinations 
are  actually  solutions  of  the  simultaneous  system  (1)  and  (2). 

This  rather  laborious  process  may  he  simplified  very  consid- 
erably by  making  use  of  the  results  of  the  following  article. 

246.   Equivalent   systems  of  simultaneous  equations.     Let 

us  denote  by  /(a;,  ^)  and  f\x^  y)  the  two  quadratic 
functions 


410     QUADKATIC  FUNCTIONS  OF  TWO  VARIABLES  [Art.  246 

(1)  f(x,  y)  =  Ax'^+2  Hxij  +  By"^  +  2ax+ 2  Fy  +  C, 

(2)  f'(x,  y)  =  A'x-2  +  2  H'xy  +  B'y^  +2a'x  +  2  F'y  +  C", 
and  let  us  put 

(3)  g(x,y)  =  a.f(x,y:,  +  b.f(x,y\ 

(4)  g'  (X,  y)  =  a'  ■  /(.r,  y}  +  h'  -fix,  y), 

where  a,  J,  a',  and  6'  are  four  constants  whose  values  may 
be  chosen  arbitrarily.  Then  g(x^  y/)  and  g'  (x,  y')  will  also 
be  quadratic  functions  of  x  and  y,  although  in  some  special 
cases  ^(a:,  ?/)  or  ^'(.r,  ^)  may  reduce  to  a  linear  function, 
^(.r,  y~)  and  ^(.r,  ^)  are  said  to  be  linear  combinations  of 
fix,  y)  and/'(.r,  ^). 

If  we  multiply  both  members  of  (3)  by  h\  those  of  (4) 
by  —  6,  add,  and  then  interchange  members,  we  find 

{ah'  -  a'b}f{x,  y)  =  b'  ■  g{x,  y)  -  b  ■  g' (x,  y}. 

Simihirly,  let  us  multiply  both  members  of  (3)  by  —  a', 
those  of  (4)  by  +  a,  and  add.     We  find 

(ab'  -  a'b^f'(x,  y)  =  -  a'  ■  g{x,  y')  +  a  -  g'Qx,  y). 

If  ab'  —  a'b  is  different  from  zero,  we  may  write  the  last 
two  equations  as  follows  ; 

(5)  fix,  ?/)  = -^^     ^/, r/-^^' 

ab  ~  a'b 

^^\  /C-^'^)-  ab'-a'b 

In  other  words,  if 

a      b 
(7) 


.'        A' 


=  ab'  —  a'b^O, 


not  only  will  g(x,  y^  and  g' (x,  ?/)  be  linear  combinations  of 
f(x,  ?/)  and  f'(x,  y}-,  but  conversely  f(x,  y')  and  f'(x,  y^  will 
also  be  linear  combinations  of  g(x,  ?/)  and  g' {x,  ?/). 

Now  let  (x,  y)  be  any  solution  of  the  simultaneous  quad- 
ratic equations 

(8)  /(^,  J/)  =  0,    f'ix,y-)  =  0. 


Art.  247]  NORMALIZATION  411 

According  to  (3)  and  (4),  (a;,  y/)  will  also  be  a  solution  of 
the  simultaneous  quadratics 

(9)  gix,n-)  =  %     g'{x,y)==Q. 

Conversely,  according  to  (5)  and  (6),  any  solution  of  (9) 
will  also  be  a  solution  of  (8).  Consequently  the  two  sys- 
tems (8)  and  (9)  have  exactly  the  same  solutions,  that  is, 
every  solution  of  one  system  is  also  a  solution  of  the  other. 
In  other  words,  the  two  systems  are  equivalent.  We  have 
proved  the  following  important  theorem  : 

All   of    the    solutions   of    a    system    of   two    simultaneous 
quadratics 

(10)  /(.r,2/)  =  0,    /'(.r,i/)  =  0 
are  also  solutions  of  the  system 

(11)  a  .f(x,  y)+h  ./(.T,  y)=0,  a'  .f(x,  y}+b'  .f'(x,  y)  =  0, 

whose  left  members  are  said  to  he  linear  combinations  with  con- 
stant coefficients  a,  b,  a',  ?/,  of  the  left  members  of  (10).  More- 
over, the  tivo  systems  (10)  and  (11)  are  equivalent,  if  the 
constant  coefficients  a,  b,  a\  b'  are  such  as  to  make 


(12) 


a      b 
a'     b' 


=  ab'  -  a'b  =^  0. 


247.  Normalization.  We  shall  now  make  use  of  this 
theorem  to  simplify  the  solution  of  a  system  of  simultaneous 
quadratic  equations 

(1)  f(x,  y)  =  Ax^  +2Hxy  +  By'^  +  2ax+2Fy  +  C=0, 

(2)  f'{x,  y-)  =  A'x^  +  2H'xy  +  B' y'^+2a'x+2 F'y  +  C  =Q. 

Let  us  make  the  following  linear  combinations  of  f(x,  y') 
aud/(.c,  ^): 

(3)  gir,  y^  =  B'  .fix,  y)-B  -fix,  y), 
g'Qc,  y)  =  -A'  -fQx,  y^)  +  A  -fix,  y). 


412     QUADRATIC  FUNCTIONS  OF  TWO  VARIABLES  [Art.  247 

In  the  notation  of  Art.  246,  this  amonnts  to  putting 

a=B',h  =  -B,a'  =  -A',h'  =  +  A, 

so  that  we  have,  in  our  case, 

ah'  -a'h  =  AB'  -A'B. 

According  to  the  final  theorem  of  Art.  246,  the  equations 
g(x,  3/)  =  0  and  g' (x^  ^)  =  0  will  therefore  form  a  system 
equivalent  to  (1)  and  (2)  if 

(4)  AB'-A'B^O. 

But  we  find,  from  (1),  (2),  and  (3),  that  these  equations 
become 

(5)  c/{x,  g)  =  (AB'-A'B).i^-\-2(HB'-H'B)xi/+^  +  -  =  0, 
and 

(6)  g'{x,y)=^-{-2(AII'-A'Hyxi/+(AB'-A'B}f-h"-  =  0, 

where  we  have  merely  indicated  the  terms  of  lower  degree 
than  the  second  by  dots,  and  where  the  asterisk  >|<  indicates 
that  the  corresponding  term  has  not  been  omitted  by  mis- 
take, but  that  the  coefficient  of  that  term  is  actually  equal 
to  zero. 

We  may  rewrite  equations  (5)  and  (6)  as  follows : 

"^  ^       ^  +  2H^xy  +  B^y'^+2a^x+2F^ij+  C^  =  0, 

if  we  put 

A^  =  AB'  -  A'B,  H^  =  HB'  -  H'B,  etc. 

Thus,  we  may  replace  system  (1),  (2),  by  a  system  equiva- 
lent to  it  of  the  simpler  form  (7)  provided  that  AB'  —  A'B 
is  not  equal  to  zero. 

Let  us  consider  now  the  case,  hitherto  excluded,  that 

(8)  AB'  -  A'B  =  0. 

Then  equation  (6)  contains  at  most  one  second  degree  term, 
namely 

(9)  2(iAH'  -  A'H^xy  =  2  H^xy, 


Art.  247]  NORiMALIZATION  413 

siuce  the  coefficient  of  a-^  is  equal  to  zero,  on  account  of  (8). 
It  may  happen  tliat  the  term  (9)  is  also  absent,  namely  if 
H^  =  AH'  —  A'R=  0.  In  that  case  the  equation  g'(x, «/)  =  0 
becomes  linear,*  and  our  system  of  equations  may  be  solved 
by  the  method  of  Art.  244.  If,  however,  H^  =^  0,  we  may 
replace  the  two  equations  (1)  and  (2)  by  tlie  equivalent 
system 

^2^+2  Hxy  +  By^+'iax  +  2Fy+  O'  =  0 
*  +  2  K^y  +  *  +2  (^..r  +  2  F^_y  +  C^  =  0, 

and,  by  means  of  the  second  equation,  we  can  eliminate  the 
xy  term  from  the  first,  giving  rise  to  a  new  equivalent  sys- 
tem of  the  form 

^..         A.x''  +  *  +  By-  +  2  a,x  +  2F^y+  C\  =  0, 
'^     ^  *  +  2  ff^xy  +  *  +  2  (7,.r  +  2  F,y  +  (7.  =  0. 

Thus,  by  the  process  of  linear  combinations,  we  may  always 
reduce  a  system  of  two  independent  simultayieous  quadratics  tvith 
two  unknowns  either  to  the  form  (7),  or  to  the  form  (10),  or 
else  to  a  system  in  which  one  of  the  equations  is  of  the  first 
degree  only. 

When  a  system  of  simultaneous  quadratics  has  been  re- 
duced to  one  of  these  forms,  we  shall  say  that  the  S3'stem 
has  been  normalized. 

One  great  advantage  of  this  normalization  consists  in  the 
greater  facility  which  it  gives  us  in  making  a  graphic 
solution.  The  loci  of  both  of  the  equations  of  the  normal- 
ized system  (7)  are  easily  plotted  because  one  of  these 
equations  gives  y  as  a  rational  function  of  x,  while  the  other 
gives  X  as  n,  rational  function  of  y.  These  graphs  fall  under 
the  case  2  of  Art.  243.  If  the  normalized  system  is  of  form 
(10),  the  second  equation  will  again  give   y  as  a  rational 

*  It  might  even  happen  that  all  of  the  coefticiciirs  of  this  equation  are  equal 
to  zero.  In  that  case  we  say  that  theoriijinal  two  equations  are  not  independent; 
one  of  them  is  a  mere  multiple  of  the  other,  the  graphs  of  the  two  equations 
coincide,  and  every  solution  of  one  equation  satisfies  them  hoth.  We  are  assum- 
ing tacitly  that  the  two  given  equations  are  independent,  so  that  this  case  may 
be  I'egarded  as  excluded. 


414     QUADRATIC  FUNCTIONS  OF  TWO  VARIABLES  [Art.  248 

function  of  x.  The  graph  of  the  first  equation  will  also  be 
obtained  easily,  although  it  does  not  determine  ^  as  a 
rational  function  of  x,  because  this  graph  comes  under 
Case  4  of  Art.  243. 

The  student  should  not  make  the  mistake  of  thinking  that 
the  graphs  of  the  two  original  equations  are  the  same  as 
those  of  the  two  equations  obtained  from  them  by  normaliza- 
tion. They  are,  in  general,  quite  different,  and  it  is  pre- 
cisely this  difference  which  accounts  for  the  fact  that  the 
graphs  are  easier  to  construct  for  the  normalized  system 
than  for  the  original.  However,  the  points  of  intersection  of 
the  two  pairs  of  graphs  are  the  same,  and  that  is  the  only 
thing  we  care  about  just  now. 

EXERCISE  CXIV 

Normalize  the  following  systems  of  quadratics  : 
1.    X-  +  xy  +  if-  -  X  -  .?/+  2  =  0,  3.  2  x-  +  Z  xy-\-if-\-1x-  5  ,y+  1  =  0, 

.r^  —  3  zy  —  _y-  +  x  +  //  —  3  =  0.  6  x^  +  9  x^  +  3  y/^  +  x  -f  ?/  + 1  =  0. 

2    x2  + XT/ +  ?/'^  +  3x -4  ?/ +  7  =  0,       4.  2x-  +  4xir-3  ?/2  +  2x-5?/4-l  =  0 

2  x^  -  xy  +  2  y'^  +  X  +  y  -  2  -  0.  o  x'^-oxy  +  2y^-x+  7  ?/+  2  =  0. 

248.  Existence  of  four  solutions.  Let  us  assume  that 
AB'  —  A'B  is  different  from  zero,  and  let  us  normalize  our 
system.  We  may  then  consider  the  normalized  system  (7) 
of  Art.  247  instead  of  the  original  two  equations.  PVom 
the  second  of  these  equations  we  find,  solving  for  a:, 

so  that  a;  is  a  ratiotial  function  of  y.  If  we  substitute  this 
value  of  X  in  the  first  of  the  equations  (7),  Art.  247,  we  find 
the  following  fractional  equation  for  y  : 

,  f  ^^2  +  2  F,^  +  C,r-     jj-    B,f^  +  ^.  F4,  +  (7, 

_  ^^^2^!+lZ^±^  +  2  #,^  +  (7,  =  0, 
or,  clearing  of  fractions. 


Art.  2i8]         EXISTENCE   OF   FOUR  SOLUTIONS  415 

-  4  H,y<iH^y  +  a,)(B,y^  + '2  F,y  +  C^) 

-  4  a,iff,y  +  G^)(B^jf  +  -2F,y+  C^) 
+  KH^y+  a^)\^F,y+C\)  =  0. 

If  we  perform  the  indicated  multiplications,  a  rather  long 
but  perfectly  elementary  operation,  we  observe  that  (2)  is 
an  equation  of  the  fourth  degree  in  y,  unless  the  coefficient 
of  y^  should  happen  to  be  equal  to  zero.  Thus  there  will 
exist,  in  general,  four  numbers  t/j,  j/g'  ^3'  ^4'  which  satisfy 
equation  (2).  (See  Art.  126.)  If  we  substitute  in  order 
these  four  values  of  y  in  (1),  we  find  four  corresponding 
values  of  a:,  which  may  be  called  x^^  a-g,  Xg,  x^.  We  thus  find 
four  solutions,  namely,  (a-j,  y{),  (x^,  y^),  (x^,  yg),  and  (a:^,  y^) 
of  the  normalized  system,  and  therefore  also  of  the  original 
system. 

If  AB'  —  A'B  is  equal  to  zero,  the  normalized  system  has 
the  form  (10),  Art.  247,  instead  of  (7),  Art.  247,  but  the 
argument  remains  essentially  the  same.  We  have,  therefore, 
the  following  result. 

A  system  of  two  simultaneous  quadratics  with  two  unknowns 
in  general  has  four  solutions. 

Of  course,  since  these  solutions  may  be  obtained  by  first 
solving  an  equation  of  the  fourth  degree  with  a  single  un- 
known, and  since  the  roots  of  such  an  equation  may  be  real 
or  complex,  finite  or  infinite,  distinct  or  coincident  (see 
Arts.  126  and  134),  we  are  not  asserting  that  the  four  solu- 
tions of  the  system  are  all  real,  finite,  and  distinct.  We 
may  say,  however,  that  two  simultaneous  quadratics  with  tu'o 
unknowns  have  at  most  four  distinct  real,  finite  solutions  if  the 
two  equations  are  independent.  (See  footnote  on  page  413 
for  the  definition  of  independence.) 

The  graph  of  each  of  the  two  equations  is  a  conic,  and 
these  two  conies  do  not  coincide  if  the  equations  are  inde- 
pendent. We  have  therefore  given  an  algebraic  proof  of 
the  following  theorem: 


416     QUADRATIC  FUNCTIONS  OF  TWO  VARIABLES  [Art.  249 

Two  distinct  conies  have  four  points  of  intersection.  Since, 
however,  some  of  these  points  of  intersection  may  have 
imaginary  or  infinite  coordinates,  and  since  several  of  them 
may  coincide,  we  may  state  this  theorem  somewhat  more 
concretely  by  saying  that  two  distinct  conies  never  have  more 
than  four  real  and  distinct  points  of  intersection. 

249.  Special  cases  of  two  simultaneous  quadratics.     As  we 

have  just  seen,  the  solution  of  two  simultaneous  quadratics 
may  always  be  reduced  to  tlie  problem  of  finding  the  roots 
of  a  certain  equation  of  the  fourth  order  with  a  single  un- 
known. But  in  some  cases  this  equation  of  the  fourth  order 
reduces  to  a  quadratic,  and  in  some  other  cases  it  may  be 
solved  by  the  methods  which  are  used  in  solving  quadratics. 
The  following  four  articles  are  devoted  to  the  discussion  of 
some  such  cases. 

250.  Case  I.  H=H'  =  F=F'  =  G  =  G'  =  0.  Neither  equa- 
tion contains  a  first  degree  term  nor  a  term  in  xy.  The  equa- 
tions are  of  the  form 

(1)  Ax'^  +  Bf+C=0, 

A'aP'  +  B'y-^  +  C'=0. 

These  equations  may  be  regarded  as  linear  equations  with 
ofi  and  y^  as  unknowns.  Solve  them  for  x^  and  y"^^  and  then 
obtain  x  and  y  by  extracting  square  roots. 

This  method  of  sohition  is  quite  in  accord  with  the  spirit  of  our 
s^eneral  method.  To  solve  (1)  for  x^  and  //-  is  essentially  the  same  thing 
as  to  normalize  system  (1)  by  the  first  method  of  Art.  247.  The  equa- 
lion  ot  the  fourth  degree  (2)  of  Art.  248  in  this  particular  case  reduces 
lo  a  pure  quadratic. 

EXERCISE  CXV 
1.    Find  the  solutions  of  the  system 

\Qx'^  +  21y''-  576  ^  0, 


^^^  I  a;2  +  7/2  _  25  =  0, 

and  illustrate  by  graphs. 

Solution.     From  (1)  we  find 

(2)  :c2  -  9  =  0,  2/2  -  16  =  0, 


Arts.  250,  251]        NO   FIRST   DEGREE   TERM 


417 


and  system  (2)  is  equivalent  to  (1). 
find 


(See  Art.  24G.)     Consequently  we 


H' 

+1/ 

H 

1 

D 

L 

^ 

J 

— 1 

— 

i; 

is 

& 

Jf 

/ 

/ 

\ 

y 

^ 

\ 

- 

/ 

\ 

\ 

A' 

c, 

-■1 

\ 

y 

\ 

^^ 

/. 

/ 

, 

s 

K 

B 

A 

/ 

.v 

s 

•^ 

=^r 

>~. 

F 

h 

K 

(3)  a,-  =  ±  3,  y  =  ±  4, 
giving  the  four  solutions, 

(4)  (4-3,  +4),  (-3,  +4),  (-3,  -4),  (+3,  -4).- 

The  graj)!!  of  the  first  ecjuation  of  (1)  is  the  ellipse  ABA'B',  that  of 
the  second  equation  (1)  is  tlie  circle  CDEF  oi  Fig.  74.  These  intersect 
in  the  four  points  Q,  R,  S,  T  whose 
coordinates  are  given  by  the  four 
solutions  (4). 

But  observe  how  much  easier 
it  is  to  draw  the  graphs  of  the 
equations  (2).  The  graph  of  the 
first  equation  (2)  consists  of  the 
two  lines  IIK  and  H' K',  that  of 
the  second  equation  (2)  consists  of 
LM  and  L'M'.  These  two  pairs  of 
lines  intersect  in  the  same  four 
points  Q,  R,  S,  T  as  the  ellipse  and 
the  circle,  as  they  should  according  to  the  general  theory,  since  systems 
(1)  and  (2)  are  equivalent. 

Solve  the  following  systems  algebraically  and  graphically.  In  Ex- 
amples 6-9  discuss  also  the  conditions  under  which  the  solutions  obtained 
will   be  real  and  distinct,  coincident,  or  imaginary. 

2.  X-  +  //2  =  113,  ^    £:  !_  .?^  -  1 

X2  _   ,y2  =15.  ■      a-2         fy^ 

3.  9  x2  +  2.5  f  =  225,  2,-2  +  v/2  ^  r^ 
x2  +  3/2  =  16. 

4.  9  x2  +  25  2/2  =  225, 
X-  +  ?/2  =  9. 

5.  9x2  +  25  7/2  =  225, 
x2  +  f  =  4. 

6.  ./■2  +  //2  =  k; 
x-2  -  y/2  =.  I. 


Fig.  74. 


8. 

a2 

- 

i2 

1, 

x2 

+ 

//-  = 

,.2 

9. 

X- 

+ 

/>2 

1, 

X-2 
«2 

- 

/;2 

1. 

251.  Case  II.  F=F'=G=G'  =  0.  This  case  includes 
the  preceding  one  and  is  more  general.  Neither  equation  con- 
tains any  first  degree  terms,  but  either  or  both  equations  may 
contain  an  xy  term.     The  equations  are  of  the  form ; 


418     QUADRATIC  FUNCTIONS  OF  TWO  VARIABLES  [Art.  251 

,. .  f  A.r^  4-  2  Ht?/  +  Bf+C=0, 

^  ^  \a'x'^+  2  R't>/  +  B'f  +  C  =  0. 

By  linear  combination  we  can  obtain  from  these,  the  equation 

C QAx^  +  2  Rri/  +  %2  +  C-)  -  C(A 'x^  +2  R'xy-\-B'f+ C')  =  0 

or 

(2)  (iAC -A'  C)x^-  +  2(HC'  -  IT'  C)xy  +  {BC'~B'  0)^=0, 

which  is  homogeneous  in  x  and  y  and  whose  graph  therefore 
consists  of  a  pair  of  straight  lines  through  the  origin.  (See 
Art.  240.)  The  system  composed  of  (2)  and  either  one  of 
the  original  equations  is  equivalent  to  (1).  But  (2)  may 
be  decomposed  into  two  separate  equations  of  the  form 

(3)  ax  +  hy=0,  {■\~)  a' x  +  h' y  =  0, 

if  ax  +  hy  and  a' x  +  h'y  are  the  factors  of  the  left  member  of 
(2).      Hence  by  solving  each  of  the  first  degree  equations 

(3)  and  (4)  simultaneously  with  one  of  the  equations  (1) 
Ave  obtain  the  four  solutions  of  (1). 

Illustrative  Example.     Solve  the  following  system  of  equations: 

(4)  a;-  +  3  xy  -  28  =  0. 

(5)  x2  +  f-  20  =  0. 

Solution.  To  form  the  homogeneous  equation  (2)  in  this  case,  we 
multiply  both  members  of  (4)  by  —  .5,  those  of  (5)  by  7  and  add.  This 
gives 

2x--  loa-// +  7//2  =  0, 
or,  factoring, 

(6)  (2x-y){x-ly)=0. 

The  system  composed  of  (5)  and  (6)  is  equivalent  to  the  original  system 
(4),  (5).     But  according  to  (6)  we  have  either 

(7)  y  =  2  X,     or     (8)  y  =  ^  X. 
Substituting  the  value  (7)  in  (5)  gives 

^2  +  4  x2  =  20,   5  .r2  =  20,    .r2  =  4,   x=±  2, 

and  therefore  from  (7),  y  =±i.  Thus  (+2,  +4)  and  (-2,  -  4)  are 
two  solutions.  Substituting  the  value  (8)  in  (.5)  gives  the  other  two 
solutions  (+|\/10,  +  ^VlO)  and  (-^VIO,  -|VlO). 


Art.  251] 


NO   FIRST  DEGREE   TERM 


419 


A  second  solution.  The  following  modified  form  of  the  solution  is 
convenient.  Since  we  know  that  by  finding  the  factors  of  the  homo- 
geneous function  which  occurs  on  the  left  member  of  (2),  and  equating 
one  of  these  factors  to  zero,  we  shall  obtain  an  equation  of  the  first 
degree  of  the  form  y  =  7nx  (see  (7)  and  (8)),  we  substitute  //  =  mx  in  the 
given  equations,  and  regard  m  as  an  unknown  quantity.  Thus  (4)  and 
(.5)  become 

a;2  +  3  nix^  =  28,   x-  +  m-x-  =  20, 

Solving  each  of  these  equations  for  x-  gives 

28  20 


1  +  3  7«      1  +  m^ 
whence  a  quadratic  equation  for  //(,  namely 

7  7n2  -lo?H  +  2=0, 

m  =  2  or  +. 


■whose  roots  are 


Since  we  had  piit  //  =  ?nx,  we  now  know  that  either 

?/  =  2  .r  or  ?/  =  f  X. 

These  are  the  same  equations  as  (7)  and  (8)  and  now  we  proceed  as  in 
the  first  solution. 

Figure  75  shows  the  graph- 
ical solution.  The  graph  of 
(4)  is  the  hyperbola  of  Fig. 
75  and  the  graph  of  (5)  is  the 
circle  shown  in  the  same  fig- 
ure.    The   coordinates  of  the 


+y 

^ 

/ 

/ 

r   7 

Y  I 

_, 

A^ 

/ 

/ 

/  \^ 

s 

^ 

/ 

/ 

^ 

n\ 

_ 

_- 

N 

/ 

_— - 

-m 

t;. 

— - 

'70 

^ 

V. 

c\ 

s 

/ 

1 

■•^ 

\ 

iy 

/ 

/ 

A/ 

OjV 

■^ 

/    ' 

\ 

/ 

1 

/ 

— 

7^ 

—  - 

— 

points  of  intersection  ^1,  B,  C, 
D  are  the  solutions  of  our  sys- 
tem. The  graph  of  (G)  is 
composed  of  the  two  straight 
lines  AC  and  BD  which  pass 
through  the  origin  0.  We 
might  find  the  solutions  graph- 
ically by  drawing  the  circle 
(the  graph  of  (5)),  and  these 
two  straight  lines,  and  then  measuring  the  coordinates  of  the  four  points 
of  intersection.     Our  algebraic  process  is  exactly  equivalent  to  this. 


Fig.  75 


EXERCISE  CXVI 
Solve  the  following  systems  algebraically,  using  the  simplest  graphs 
which  you  can  find  to  accomplish  the  purpose.     This  means,  for  instance, 
to  use  the  circle  and  the  two  straight  lines  of  Fig:  75,  rather  than  the 
cii'cle  and  the  hyperbola. 


420     QUADRATIC  FUNCTIONS  OF  TWO  VARIABLES  [Art.  252 


1. 

a;'^  4-  xy  —  y-  =  5, 

6. 

2  //'^  —  4  ^7/  +  3  X- 

=  17, 

2x^  -  oxy  +'2y'^  = 

14. 

y/2-.r-2='l6. 

2. 

x2  +  yi  =  10, 
xy  =  3. 

7. 

X2  +   ?/2  _  c[2^ 

xy  ^  k. 

3. 

x^  +  y^  =  20, 
xy  =  8. 

8. 

Z2     ,    .y^  _  J 

a2      i 

4. 

x^  +  10 //2  =  44, 

xy  =  ^•. 

5. 

xy  +  1  =  0. 

•r^  +  3  xy  =  28. 
iy^  +  xy  =  8. 

9. 

X2         .?/2  _ 

xy  =  A:. 

252.  Case  III.  Both  equations  contain  x  and  y  in  sym- 
metric fashion,  so  that  each  equation  is  left  unaltered  if  x  and 
y  are  interchanged.     The  equations  are  of  the  form 

(1)  A(x^  +  f)+2Hxi/-h2a(x+2/}+  (7=0, 

(2)  A'  (x^  +  j/2)  +  2  H'x^  +  2  a'  ix  +  y)  +  C"  =  0. 

In  this  case  it  is  advisable  to  introduce  the  fundamental 
symmetric  functions  of  x  and  y  (see  Art.  133)  as  new  un- 
knowns.    We  put  therefore 

(3)  x  +  y  =  u,   xy  =  v. 

Since 

a;2  j^y'i-^(x^yy'—2 xy, 
we  shall  have 

(4)  a;2  +  ^2  ^  w2  _  2  y, 

so  that  the  equations  (1)  and  (2)  assume  the  form 

(5)  aw^  +  5m  H-  6'v  +  c?  =  0, 

(6)  a!iC"^Vu^  c^v  +  S!  =  ^. 

From  these  two  equations  we  can  always  eliminate  ti?  by 
linear  combination,  so  as  to  obtain  an  equation  of  the  first 
degree  between  u  and  o.  If  this  first  degree  equation  be 
solved  simultaneously  with  either  (5)  or  (6)  (see  Art.  244), 


Art.  252]  BOTH   EQUATIONS   SYMMETRIC  421 

we  obtain  two  sets  of  values  (Wj,  Vj)  and  {u^,  v^)  ^^^'  ^  ^^^^^  ^• 
It  now  remains  only  to  solve  the  systems 

and 

in  each  of  which  one  of  the  equations  is  of  the  first  degree. 

ExAMPLK.     Solve  the  system 

X-  +  y^  =  25,   xy  =  12, 
by  this  method. 

Solution.     Putting  x  +  /y  =  u,  xy  =  v,  we  have 

u-2  -  2  ('  =  25,    V  =  12. 
Therefore 

u-  =  25  +  2  y  =  49,    w  =  ±  7. 

It  remains  to  solve  the  two  systems 

•^■  +  .^  =  ^'     and      l^+U  =  -7, 


xy  =  12,  [  xy  =  12. 

The  solutions  of  the  first  system  are  (3,  4)  and  (4,  3).     Those  of  the 
second  system  are  (—3,  —  4)  and  (—  4,  —  3), 

A  second  method  applicable  to  this  case  is  to  substitute 

(7)  X  =  m  +  n,   y  =  m  —  n, 
so  that 

(8)  X  +  ^  =  2  ?n,    xy  =  m?  —  n-,    x-  +  y'-^  =  2(7n-  +  n-). 

In  the  above  example  this  would  give  us 

2(m2  +  n^)  =  25,     m-  -  n^  =  12, 
whence 

4  m2  =  49,     4  n2  =  1 

TO  =  ±  I,        n  =±  |. 
and  tlierefore 

X  =  I  +  ^  =  4,  or  I  -  1  =  3,  or  -  I  +  1  =  -  3,  or  -  I  -  ^  =  -  4, 

to  be  combined  respectively  with 

y=i-i  =  -i,  or  I-(-D  =  4,  or  -  |  _  i  =  -  4,   or    _|-(-'i)  =  -3, 

giving  the  same  four  solutions  as  before. 

Geometrically,  the  symmetry  of  the  equations  (1)  and  (2) 
has  the  follow! no-  signiticance.  If  x  =  a,  y  =  h  is  a  point  on 
the  graph  of  (1),  then  the  point  x=h,   // =  a  is  also  on  the 


422     QUADRATIC  FUNCTIONS  OF  TWO  VARIABLES  [Art.  252 


graph  of  (1).  But  (see  Art.  149)  this  means  that  the  locus 
of  (1)  is  symmetrical  with  respect  to  the  line  y  =  x,  which 
bisects  the  angle  between  the  positive  a:-axis  and  y-axis. 
If  this  is  true  of  both  equations,  then  the  solutions  of  the 
system  must  correspond  to  each  other  in  pairs  in  such  a  way 
that  the  two  members  of  a  pair  belong  to  points  which  are 
symmetric  with  resj^ect  to  this  angle  bisector.  Observe  that 
this  is  actually  so  in  case  of  the  system  which  we  have  just 
solved. 

For  the  graphic  solution  of  this  case  it  is  advisable  to 
replace  (1)  and  (2)  by  an  equivalent  system  composed  of 
two  equations  one  of  which,  at  least,  shall  contain  no  xy. 
For  the  graph  of  such  an  equation  will  be  a  circle  whose 

center  lies  on  the  bisector  of 
the  angle  between  the  coordi- 
nate axes.  Sometimes  (if 
]I=  W  =  0)  both  graphs  are 
of  this  kind.  In  all  other 
cases  the  second  equation 
may  be  chosen  so  as  to  con- 
tain no  x^  +  y^  term.  Its 
graph  will  be  a  hyperbola 
with  a  vertical  asymptote  (see 
Art.  140)  unless  it  happens 
to  degenerate  into  a  pair  of 
lines,  one  parallel  to  the  rr-axis  and  one  parallel  to  the  y-axis. 
Figure  76  represents  these  graphs  for  the  system 

x^  -\-  y^  —  25,         xy  =  12 

whose  algebraic  solution  we  have  just  given. 


Fig.  76 


EXERCISE  CXVII 

Solve  the  following  systems  algebraically  and  graphically,  using  the 
simplest  graphs  which  you  can  find  to  accomplish  the  purpose : 

1.  x2  +  7/2  ^  13^  3_    ^i  +  </2  +  2(x  +  ?/)  =  11, 
u:i/  =  G.  3  xy  =  2(x  +  ?/). 

2.  X  +  y  =  30,  4.    y^  +  !f^  +  .''.y  +  ■'■  +  //  =  1", 

x)j  =  216.  x2  +  f--i  xu  +  2  .(•  +  2  2/  =  9. 


Arts.  253,  2o4]         ONE    EQUATION   COMPOSITE  423 

253.  Case  IV.  When  at  least  one  of  the  given  equations  is 
composite,  that  is,  when  its  left  member  can  be  factored  into 
two  integral  rational  functions  of  the  first  degree.  Such 
cases  ma}'  be  discovered  by  computing  the  discriminant  A 
for  the  left  member  of  each  of  the  given  equations.  (See 
Art.  240.)  If  one  of  these  discriminants  is  equal  to  zero, 
we  can  find  the  linear  factors  of  the  left  member  of  the  cor- 
responding equation  by  the  method  of  Art.  240.  If  we  find 
that  ax  +  bi/  +  c  is  one  of  these  factors,  we  solve  the  system 
composed  of  the  equation  ax  -\-  bi/  -\-  e  =  0  and  of  the  other 
quadratic  equation.  This  gives  two  solutions  (see  Art.  244). 
We  then  use  the  second  factor  in  tlie  same  way  to  obtain  the 
remaining  solutions. 

Example.     Solve  the  following  system  : 

2  x^  —  xy  —  3  //-  +  9  X  +  4  ^  +  7  =  0,     xy  =  1. 

Solution.     In  Ex.  1,  Exercise  CIX,  we  have  shown  that 

2x^-~  xy  -'dy-^  +  9x  +  iy  +  7  =(x  +  y  +  1)(2  x  -  Sy  +  7). 

Therefore  we  may  solve  separately  the  two  systems 

X  +  y  +  I  =0,     xy  =  1, 
and 

2x-Sy  +  7-0,     xy  =  1. 

The  details  of  the  solution  are  left  to  the  student. 

EXERCISE  CXVIII 
Solve  each  of  the  following  systems  by  using  an  appropriate  method. 

1.  (x  -  4 y  -  3)(r  +  t/)  =  0,  3.   x^  +  y^  =  28, 
x2  ^y-i  -  4-  xy  =  22.  X  +  y  =  i. 

2.  x2  +  y-  =  a\  Hint.     Observe  that  x  +  y  is  a. 
Vx  +  Vy  =  Va,                              factor  of  x^  +  y^. 

where  Vx,   Vy,  \  a  are   to  be  re-  4.   x*  —  y^  —  19, 

garded  as  positive  roots.  x  ~  y  =  1 . 

254.  A  new  method  for  the  general  case  of  simultaneous 
quadratics.  The  method  used  in  Case  IV,  Art.  253,  may  be 
extended  as  follows.  Even  if  the  left  member  of  neither  of 
the  two  given  equations 


424     QUADRATIC  FUNCTIONS  OF  TWO  VARIABLES  [Art.  254 

( 1)  /^(a;,  y)=Ax^  +  ^IRxy  +  %2  +  2  ax  +  2^^/  +  (7=  0, 

(2)  f^(x,  y^  =  Mx''  +  "IR^xy  +  B  y'^  +  2a'x +  'lFy+ C  =  0 

can  be  decomposed  into  a  product  of  two  linear  factors,  it  is 
always  possible  to  find  a  linear  combination  of  the  functions 
fi(x^  y^  and/2(2;,  y)  which  can  be  factored. 

To  prove  this,  we  form  the  general  linear  combination 
^/lO^;  y^+M^^  y\  of  (1)  and  (2). 

This  gives  the  new  equation 

(3)  (Ak  +  A')2^  +  liHh  +  H')xy  +  {Bk  +  B'yf 

+  2((7yt  +  a')x  +  2(Fk  +  F')y  +  Ck+C'  =  0. 

The  left  member  of  (3)  will  be  a  product  of  linear  factors, 
if  and  only  if  its  discriminant  (see  Art.  240)  is  equal  to 
zero,  that  is,  if  and  only  if 

(4)  (Ak-{-A'}(Bk  +  B'}(Ck  +  C') 

+  2(Fk  +  F')(ak  +  G')(Hk  +  H'}  -  (Ak  +  A')(Fk  +  F')^ 

-(Bk  +  B')(ak+  a'y-iCk+  o'^{Hk+  H>y  =  Q. 

Now  (4)  is  a  cubic  equation  for  k.  If  the  k  which  occurs  in 
(3)  is  one  of  the  roots  of  this  cubic  equation,  the  left  member 
of  (3)  will  be  a  product  of  two  linear  factors,  and  the  system 
composed  of  (1)  and  (3),  which  is  equivalent  to  the  original 
system,  may  be  solved  by  the  method  of  Art.  253. 

This  method  is  applicable  in  all  cases,  but  requires  the 
solution  of  the  cubic  equation  (4)  for  k.  Thus  we  see  that 
the  general  problem  of  solving  any  system  of  simtdtaneous 
quadratics  may  he  reduced  to  that  of  solving  a  certain  auxiliary 
cubic  equation  with  one  unknown.  Tliis  method  is  somewhat 
simpler  than  that  of  Art.  248,  where  we  required  the  solu- 
tion of  an  equation  of  the  fourth  degree.  However,  the 
two  methods  are  closely  related.  For,  as  we  saw  in  Art.  124, 
the  solution  of  an  equation  of  the  fourth  degree  requires  as 
a  preliminary  the  solution  of  an  auxiliary  cubic  equation. 

The  method  outlined  in  this  article  becomes  immediately 
available  in  particular,  whenever  one  of  the  roots  of  the 
cubic   equation   (4)   happens  to   be  evident  by  inspection. 


Akt.  2r)o]     THE   METHOD   OF   SMALL   CORRECTIONS  425 


Cases  II  and  IV  (Arts.  251  and  253)  are  really  illustrations 
of  this  general  method. 

EXERCISE  CXIX 
1.    Set  up  the  cubic  equation  for  k  in  llie  case  where  the  two  given 
equations  are  of  the  form 

Ax^  +  2  Hxi/  +  Bf  +  r*  =  0,  A'y-  +  2  //'./■//  +  />"//-  +  <"'  =  0, 
and  show  how  the  method  of  tliis  article  applies  to  Case  II. 

255.  The  method  of  small  corrections.  It  often  happens 
that  the  graphic  solution  gives  only  approximate  values  of  x 
and  y  and  that  none  of  the  algebraic  methods  which  we  have 
discussed  will  be  convenient  for  actual  numerical  work.  In 
such  cases  it  is  usually  desirable  to  employ  the  method  illus- 
trated in  the  following  example.  * 

Find  a  solution  of  the  simultaneous  (juadratics 

(1)  4x-'  +  !)y^=l 

(2)  {x-\y+{ii-\Y  =  i 

correct  to  six  decimal  places. 

The  graph  of  (1)  is  the  elliiise  ABA'B'  (Fig.  77)  which  has  the 
X-axis  and  ?/-axis  as  axes  of  symmetry  and  for  which  OA  —  \,  and  OB  =  \. 
The  graph  of  (2)  is  a  circle 
of  radius  unity,  whose  center 
C  has  the  coordinates  x  =  1, 
y  =  \.  Figure  77  shows  that 
one  point  of  intersection  5  is 
given  approximately  by  x  =  0, 
y  =  \,  and  the  other  point  of 
intersection  T  has  coordinates 
which  are  approximately  x  =  \, 
y  =  —  \.  We  propose  to  find 
more  exact  values  for  the  co- 
ordinates of  T. 

We  regard  the  values 
X  =  0.33,  */=— 0.2.5  as  a  first 
approximation.    Let 

(3)  X  =  0..33  +  h,  y=-  0.25  +  k 

*  This  example  has  been  taken  from  Ruxge's  Praxis  der  Gleichnncjen, 
Leipzig,  1900,  p.  73. 


+y 

+1 

/ 

^ 

/ 

B 

s 

•(1,^4) 

A 

O 

\> 

+1              / 

B' 

^-^ 

- — -^ 

-1 

Fig.  77 


426     QUADRATIC  FUNCTIONS  OF  TWO  VARIABLES  [Art.  255 

be  the  true  values  of  these  coordinates.  Then  h  and  k,  the  corrections, 
may  be  regarded  as  small  quantities  whose  squares  and  products  may  be 
neglected.  If  we  substitute  the  values  (3)  in  (1)  and  (2),  neglecting 
h',  hk,  and  ^-^,  we  find  the  equations  of  the  first  degree 

,,.  2.64  A  -  4.5 A:  =  0.0019, 

W  1.34  A  +  1.50  yL-  =  0.0114 

for  the  corrections.  Let  us  solve  these  equations  of  the  first  degree  for 
Ti  and  k,  neglecting  all  figures  beyond  the  5th  decimal  place.     We  find 

(5)  /«  =  +  0.00541,  k  =  +  0.00275. 
Substituting  these  values  in  (3)-  gives  the  values 

(6)  X  =  0.33541,  y^-  0.24725 

which  we  regard  as  a  second  approximation  to  the  desired  coordinates 
of  T. 

We  now  use  these  new  values  of  x  and  ij  and  put 

X  =  0.33541  +  h^,  y^-  0.24725  +  k^, 

where  /;,  and  k^  are  the  further  corrections.  We  obtain  the  following 
equations  for  Aj  and  Atj, 

2.68  \  -  4.45  k^  =  0.000192, 
1.33^1  +  1.49  yti  =  0.000062, 
giving 

hi  =  -  0.000001,  ^■l  =  +  0.000042. 

The  corrected  values  of  x  and  y  will  be  (to  six  decimal  places) 

X  =  0.335409,  1/  =  -  0.247208. 

As  a  check  we  substitute  these  values  in  our  original  equations.     We 

find 

4x^  +  9  >f  -  1  =  0.00000283, 

(x  -  1)2  +  (/y  —  *)•■=-  1  =  0.00000097. 

A  third  approximation  would  not  alter  our  results  by  as  much  as  a 
single  unit  of  the  sixth  decimal  place.  Therefore  the  values  obtained 
are  correct  to  six  decimal  places. 

This  method,  known  as  the  method  of  small  corrections,  is 
applicable  not  only  to  simultaneous  quadratics  with  two  un- 
knowns but  to  equations  of  higher  degree  involving  more 
than  two  unknowns.  Newton's  method  (see  Art.  97)  is 
really  a  special  case  of  the  method  of  small  corrections. 


Art.  256]  APPLICATIONS  427 

EXERCISE  CXX 
1.    Find  to  six  decimal  places  that  solution  of 

X-!  -  5  x^-;/*  +  1506  =  0, 
y5  _  ;3  -giy  _  103  =  0, 

which  is  given  approximately  hy  x  =  2,  y  =  3. 

256.     Applications  which  involve  simultaneous  quadratics. 

The  following  exercise  contains  concrete  pi'oblenis  which 
lead  to  simultaneous  quadratics.  They  have  all  been  formu- 
lated in  terms  of  general  numbers.  If  the  student  wishes 
to  obtain  specific  numerical  applications,  he  may  of  course 
substitute  particular  numbers  for  the  quantities  a,  6,  c,  etc., 
which  are  to  be  regarded  as  known  in  each  of  these  examples. 
But  the  practice  of  setting  up  the  equations  and  solving  them 
by  general  formulas  is  more  important,  at  this  stage,  than 
that  of  obtaining  numerical  results,  since  the  student  has  had 
ample  practice  in  obtaining  such  results  in  the  earlier  parts  of 
this  chapter. 

EXERCISE    CXXI 

1.  The  Slim  of  two  numbers  is  «,  and  their  product  is  b.     Find  the 
numbers. 

2.  The  area  ^  of  a  right  triangle  and  its  hypotenuse  c  are  given. 
Find  formulas  for  the  lengths  of  the  other  two  sides,  a  and  b. 

3.  The  width  of  a  circular  track  is  1/n  of  its  inside  diameter.     The 
area  of  the  track  is  A  square  feet.      Find  the  dimensions  of  the  track. 

4.  Two  circles,  whose  centers  are  on 
the  same  diameter  of  a  third  circle,  are 
tangent  to  each  other  externally  and  touch 
the  third  circle  internally.  (See  Fig.  78.) 
If  r  denotes  the  radius  of  the  outer  circle 
what  must  be  the  railii,  Tj  and  7-„,  of  the 
smaller  circles  in  order  that  the  sum  of 
their  areas  may  be  just  half  the  area  of 
the  outer  circle?  What  values  must  r^ 
and  Tj  have,  in  order  that  the  sum  of  the 
areas  of  the  small  circles  may  be  equal  to 
any  specified  fractional  part  m/n  of  the 

large  circle?  Wiiat  is  the  smallest  value  which  the  sum  of  the  areas  of 
the  two  small  circles  can  ever  have  ? 


CHAPTER    XTV 

SEQUENCES    AND    SERIES    WITH    A    FINITE    NUMBER 
OF    TERMS 

257.   Continuous  and  discontinuous  variation.     All  of  the 

questions  which  have  occupied  our  attention  so  far  were  in- 
timately connected  with  the  notion  of  a  function^  and  we 
discussed,  in  order,  tlie  principal  properties  of  linear  and 
quadratic  functions,  rational  functions  both  integral  and 
fractional,  algeln'aic  irrational  functions,  exponentials  and 
logarithms.  Finally,  in  the  last  two  chapters  we  studied 
linear  and  quadratic  functions  of  two  independent  variables. 

In  some  cases  the  function  f(x)^  which  was  under  con- 
sideration, was  defined  as  a  real  function  for  all  real  values 
of  X.  In  other  cases  the  function  was  defined  only  for  a 
certain  range  of  values.  For  instance,  the  functions  log^a; 
and  -\/x  were  defined,  as  real  functions  of  x,  only  for  posi- 
tive values  of  x^  and  the  function  Vl  —  a^  is  defined  as  a 
real  function  only  for  values  of  x  between  —  1  and  +  1. 
But  in  all  cases  we  admitted  that  x  might  assume  any  one 
of  the  values  of  the  range  for  which  the  function  was  defined. 
We  always  thought  of  x  as  changing  its  value  gradually^ 
from  its  initial  to  its  final  value,  without  omitting  any  of 
the  intermediate  values.  In  other  words  ive  have  so  far^ 
almost  exdusively^  thought  of  x  as  varying  continuously. 

We  now  propose  to  think  of  x  as  varying  discontinuously. 
More  specifically,  we  shall  allow  x  to  assume  only  integral 
values.  Moreover,  we  shall  usually  confine  ourselves  to 
positive  integral  values  of  x. 

Although  we  are  only  now  beginning  to  enter  upon  a  systematic  study 
of  functions  of  x  when  x  is  confined  to  integral  values,  we  have  considered 
a  few  especially  important  cases  of  this  kind  earlier  in  our  course.  (Com- 
pare Art.  .'36  on  arithmetical  progression,  also  Art.  163,  Theorem  X.) 

428 


Art.  2r,8]  DKFIXmoX    OK    A    SKQUENCE  429 

258.  Definition  of  a  sequence.  Let  us  consider  the  values 
which  a  function /'(•'■)  iissumes  when  x  assumes  in  succession 
tlie  vahies  1,  2,  3,  4,  ■•■,  n.     These  values  will  he  ecjual  to 

/(I),  ./•(•2),  ./•(;{),   ...,  /•(//). 

respectively.  Let  us  call  these  values  Wp  w.^,  u^  •■•,  ?/.„,  so 
that 

(1)  Wi  =/(-!),    W2=/(2),    ...,   7^.=/(/0- 

We  naturally  think  of  the  numbers  ?/p  u^^  •••  u„  obtained  in 
this  way  as  being  arranged  in  order ;  Wj  first,  u^  second,  Wg 
third,  •••,  iin  in  the  ?ith  place,  and  we  shall  say  that  they  form 
a  sequence,  in  accordance  with  the  following  definition  : 

The  91  umbers  of  a  set  are  said  to  form  a  sequence  if  the  in- 
divii/ual  numhers  are  regarded  as  standing  in  a  definite  rela- 
tion of  order  with  respect  to  each  other^  so  that  we  may  in  a 
perfectly  definite  manner  speak  of  the  first,  the  second,  •••,  the 
nth  number  of  the  set. 

The  kth  number  of  the  set  (1),  or  the  kth  term  of  the 
sequence  (1),  is 

(2)  %=.f(^). 

Since  k  may  assume  any  one  of  the  values  1,  2,  3,  •••,  n,  the 
expression  (2)  may  be  regarded  as  representing  any  one  of 
the  terms  of  the  sequence.  For  this  reason  we  also  speak 
of  u,,  as  the  general  term  of  the  sequence  (1). 

In  the  sequence  1,  2,  o,  •••  n,  we  have 

w,  =  1,    »o  =  2,    W3  =  8,    •■■,Uk  =  k,    •••,  w„  =  71. 
In  this  case/(x-)  =  x.     In  the  sequence  1/2,  1/4,  1/8,  ••■,  1/2",  we  have 

Mi  =  l,   M„  =  — ,   M„  =  — ,  •■•,   «i=  — »  •••»  "»=  ?r ' '^'^d /( j)  =  —  • 
2  2^  2*  2*  2"  .'  V  '      2x 

We  cannot  always  write  down  an  explicit  formula  of  the 
form  (2)  for  the  ^th  terra  of  a  sequence,  even  when  it  is 
clear  that  the  sequence  is  perfectly  well  defined. 


480  SEQUENCES   AND   SERIES  [Art.  259 

Thus,  let  ^^  ^  j^^   ^^^  ^  J  ^j^   ^^  ^  j_^^4^  g^^^ 

and  for  any  value  of  k  let  Hk  be  the  largest  decimal  fraction  with  k  digits 
to  the  right  of  the  decimal  point  whose  square  is  less  than  2.  In  other 
words,  let  ui,  mo,  us,  ••■,  U).  be  that  sequence  of  decimal  fractions  which 
appears  when  we  seek  the  approximate  value  of  ^2  to  1,  2,  3,  •••  k  deci- 
mal places.  Although  u^  is  perfectly  well  defined  by  this  statement,  we 
cannot  write  down  a  general  explicit  formula  for  m^- 

We  shall,  nevertheless,  be  able  to  say  in  all  cases,  that  the 
kth  term  of  a  sequence  is  a  function  of  k,  and  ivrite 

Uk=f(k). 

This  equation  merely  means  that  the  ^th  term  of  the  se- 
quence depends  upon  k  for  its  value,  whether  we  can  find  an 
explicit  formula  for  it  or  not.  We  shall  confine  most  of  our 
discussion,  however,  to  the  cases  where  such  an  explicit 
formula  is  given,  and  we  miglit  begin  with  the  simplest 
case  when  the  function  f  (x^  is  an  integral  rational  function 
of  the  first  order.  However,  this  case  has  been  treated 
already  and  leads  to  the  theory  of  arithmetic  progressions. 
(See  Art.  56.) 

259.  Higher  progressions.  Let  us  study  the  case  of  those 
sequences  which  naturally  come  next,  that  is,  the  sequences 
which  are  obtained  from  a  quadratic  function 

ax^  -f  hx  +  c 

when  we  substitute  for  x  in  order  the  values  1,  2,  3,  •••  n. 
The  kth.  term  of  such  a  progression  will  be 

(1)  %  =  aP  +  hk  +  c, 

and  w^_^  =  a(k  -  1)2  +  h(k  -!)+(? 

will  be  the  (k  —  l)th  term.  If  we  denote  by  A'^_j  the  dif- 
ference between  these  two  terms,  we  find 

A'^._j  =  %.  -  Uk-^  =  aB  +  hk+c-  la{B-'2  /c  +  1) -f-  h(k-  1) -f  e] 
=  ak'^-\-hk  +  c—[^ak'^+{h—  2  d)k  -\- a  -/>  +  e], 


Art.  259]  HIGHER   PROGRESSIONS  431 

whence 

(2)  A',_i  =  u,  -  n,_,  =  2ak-a  +  h. 

Since  the  right  member  of  (2)  is  of  the  £rst  degree  in  k, 

the  differences  ^ ,     , ,      ,  , 

A       A       A        ...     A' 

form  an  arithmetical  progression  (See  end  of  Art.  258),  and 
the  differences  of  these  differences  (called  second  differences^ 
are  all  equal  to  each  other.  In  fact,  if  we  denote  these 
second  differences  by  A";i_j,  so  that 

AVi=A',-AVi, 
we  find,  from  (2), 

(3)  A",_i  =  -2a{k  +  1^  -  a  +h  ~  \^ak  -  a  +  h^=2  a, 

and  the  third  differences  are  all  equal  to  zero. 

There  is  no  difficulty  in  generalizing  this  interesting 
result.  If  the  kth  term  of  a  sequence  is  expressible  as  an 
integral  rational  function  of  k,  and  if  this  function  is  of  the  nth 
degree^  so  that 

u^  =  ak""  +  hk"-'^  +  r-A;"-2_|_  ...  4.  /^  _|_  ,^^ 

the  terms  of  the  sequence  which  is  formed  by  the  nth  differences 
tvill  all  be  equal  to  each  other.  In  other  ivords,  the  nth  differ- 
ences are  constant^  that  is,  their  value  does  not  depend  upon  k. 
TJie  (n  +  V)th  and  all  differences  of  a  higher  order  will  be  equal 
to  zero. 

This  theorem  has  a  very  important  application.  When 
the  numerical  values  of  a  function  f(x)  have  been  tabulated 
for  certain  equidistant  values  of  x,  as  in  the  case  of  a  table 
of  logarithms,  the  values  of  /(.f)  which  correspond  to  inter- 
mediate values  of  x  must  be  obtained  by  interpolation.  (See 
Art.  171.)  In  the  ordinary  logarithmic  tables  the  values  of 
a;,  for  wliicli  the  function  is  tabulated,  are  so  close  to  each 
other  that  the  ordinary  method  of  interpolation,  in  which 
only  first  differences  are  used,  is  satisfactory.  In  fact  the 
second  differences  are  either  zero,  or  else  their  values  do  not 
exceed  a  single  unit  of  the  last  decimal  place  which  is  used 


432  SEQUENCES   AND   SERIES  [Art.  260 

in  the  calculation.  But  in  many  cases  it  is  impracticable  to 
make  the  tables  so  extensive.  The  Nautical  Almanac,  for 
instance,  gives  the  declination  of  the  sun,  that  is,  the  dis- 
tance in  degrees  of  its  center  from  the  celestial  equator,  for 
noon  of  every  day  of  the  year.  If  we  wish  to  find  the 
declination  of  the  sun  at  6  A.M.,  we  must  interpolate  and,  in 
this  case,  the  first  differences  are  not  approximately  constant, 
so  that  the  ordinary  method  of  inter[)olation  does  not  suffice. 
The  method  of  interpolation  by  higher  differences  rests  upon 
the  fact  that  the  converse  of  tlie  above  theorem  can  be 
established.  That  is,  if  the  nth  differences  of  a  function  are 
constant,  the  function  may  he  regarded  as  an  integral  rational 
function  of  the  nth  order.  For  a  proof  of  this  theorem,  and 
for  further  details  in  the  theory  of  interpolation,  we  must 
refer  to  other  sources. 

260.  Geometric  progressions.  Another  important  sequence 
is  obtained  by  considering  the  values  which  an  exponential 
function  ar""  assumes  for  a:  =  0,  1,  2,  3,  •••  w.  These  values, 
namely,  a,  ar,  ar"^,  ar^^  •••,  «r",  form  a  geometric  progression, 
and  we  have  already  had  tlie  opportunity  of  becoming  ac- 
quainted with  the  principal  properties  of  such  progressions. 
(See  Arts.  59-63.) 

EXERCISE  CXXII 

Write  down  the  first  four  terms  (corresponding  to  x  =  0,  1,  2,  3)  and 
the  general  term  of  the  sequences  defined  by  each  of  the  following  func- 
tions : 

1.    /^+  1.  5.    — i— .  9.    2*. 


-  '  +  2.  6.    -^ .  10.    2 


3.    x2  +  3  X 


1 

+ 
1 

x^ 

X- 

'  - 

X 

X- 

— 

1 

X- 

+ 
1 

1 

11.    3''+''5^-i. 


4.    J—.  8.    -J—.  12.    log  (3-). 

x  +  1  .r^'  +  1 

13.  Write  down  eight  terms  of  the  sequence  defined  by  the  function 
X-  +  3  .r  —  1,  and  the  first  and  second  differences  of  this  sequence. 

14.  Prove  that  the  sequence  defined  by  the  function  x^  has  the  prop- 
arty  that  its  fovn-th  differences  are  all  equal  to  zero. 


Akts.  261,  262]  SUMMATION   OF   SERIES  433 

261.  Series.  Whenever  we  have  a  sequence  Mj,  u^,  W3,  •••  m„, 
the  sum  of  all  of  the  terms  of  this  sequence, 

S„=Ui+  u^+  ■■■  +  u„, 

is  called  a  series.  Since  we  may  think  of  ;t,  the  number  of 
terms,  as  being  large  or  small,  and  since  the  sum  of  n  terms 
of  a  sequence  will  depend  upon  the  value  of  /i,  we  may  say 
that  jSn  is  a  function  of  n,  and  this  fact  is  indicated  in  the 
notation. 

Tn  the  case  of  arithmetic  and  geometric  progressions 
we  have  been  able  to  find  simple  formulas  for  S^  (see 
Art.  56  and  Art.  60),  which  make  it  possible  to  compute  the 
value  of  *S'„  without  actually  adding  up  all  of  the  individual 
terms.  Whenever  such  a  formula  has  been  found,  which 
gives  explicitly  the  value  of  >S'„  as  a  function  of  ?i,  we  say 
that  the  series  u^  +  U2+  — h  «„  has  been  summed:  the  prob- 
lem of  finding  such  a  formula  is  known  as  the  problem  of 
summation  of  series. 

262.  Summation  of  series  by  mathematical  induction.  In 
very  many  cases  we  can  obtain  a  formula  for  the  sum  of  a 
series  by  the  method  of  mathematical  induction.  In  fact  the 
summation  of  series  by  this  method  is  one  of  the  best  means 
for  becoming  thoroughly  familiar  with  the  method  of  mathe- 
matical induction,  which  is  one  of  the  most  important  forms 
of  mathematical  reasoning,  and  Avhich  we  have  used  several 
times  in  this  book  (See  Arts.  84  and  88.)  The  following 
example  will  illustrate  the  method. 

Illustrativk  Exampi,k.     We  observe  that 

(1)       1+3  =  4  =  22,    1  +  3  +  .■)  =  0  =  3'^    1  +  3  +  5  +  7  =  16  =  42. 

These  equations  suggest  that  the  following  law  may  be  true ;  the  sum  of 
the  n  first  odd  integers  is  equal  to  ur.  In  fact,  equations  (1)  prove  that 
tlie  law  is  true  in  the  three  special  cases  when  n  —  2.  3,  or  4. 

To  prove  that  the  law  actually  is  true  for  all  values  of  n.  we  first 
prove  the  following  leinina,  or  auxiliary  theorem. 

Lkmm A.  If  the  sum  of  the  first  k  odd  integers  is  equal  to  k-,  then  the 
sum  of  the  first  k  +  I  odd  integers  will  be  equal  to  (k  +  l)^. 


434  SEQUENCES   AND   SERIES  [Art.  203 

Proof  of  the  lemma.  The  kt\\  odd  integer  is  2  A  —  1,  since  the  odd 
integers  form  an  arithmetic  progression  whose  first  term  is  1  and  wliose 
constant  difference  is  2.  Therefore,  //'the  sum  of  the  first  k  odd  integers 
is  equal  to  k'^,  we  have 

(2)  1  +  3  +  5+  ...  +(2A--l)=A---2. 

The  sum  of  the  first  k  +  1  odd  integers  will  be  obtained  by  adding  the 
{k  +  l)th  odd  integer  (which  is  2  k  -\-  1)  to  the  sum  of  the  first  k  odd 
integers.     If  the  latter  is  given  by  (2),  we  shall  therefore  have 

(3)  1  +  .3  +  .0+  ...  +(2/.-l)+(2/.+  1)  =  B  +  2k  +  l  =  {k+  \y. 

In  other  words :  if  (2)  is  true,  then  (8)  must  also  be  true,  thus  proving 
the  lemma. 

But  we  know  from  (1)  tliat  the  sum  of  the  fii-st  four  odd  integers  is 
42.  Apply  the  lemma  for  the  case  k  —  4.  The  lemma  tells  us  that  the 
sum  of  the  first  five  odd  integers  is  5-.  Apply  the  lemma  to  the  case 
k  =  5.  It  tells  us  that  the  sum  of  the  first  six  odd  integers  is  6^.  Pro- 
ceeding in  this  way  we  finally  conclude  that  the  sum  of  the  first  n  odd 
integers  actually  is  equal  to  n^. 

263.  General  characteristics  of  the  method  of  mathematical 
induction.  The  theorems  which  are  capable  of  proof  by 
mathematical  induction  have  the  following  characteristics : 

(a)  The  theorem  asserts  that  a  certain  property  or  relation 
is  valid  in  all  of  a  certain  well-defined  set  of  cases. 

(6)  The  cases  with  which  the  theorem  is  concerned  can  be 
arranged  in  a  definite  order,  so  that  there  is  a  first,  a  second, 
•  ••,  a  kih.  case. 

(c)  The  total  number  of  cases  may  be  large  or  small.  But 
the  most  useful  applications  of  the  method  of  mathematical 
induction  to  theorems  of  this  sort  are  those  in  which  the 
number  of  cases  is  large. 

The  proof  of  a  theorem  of  this  kind  by  mathematical 
induction  always  consists  of  two  parts. 

The  first  part  of  the  proof  consists  in  verifying  that  the 
theorem  states  the  truth  in  the  first  few  cases.  We  shall 
call  this  part  of  the  proof,  the  verification. 

For  the  purposes  of  the  proof  by  mathematical  induction,  it  really 
suffices  to  verify  the  correctness  of  the  theorem  for  the  very  first  case. 
But  frequently,  as  in  the  illustrative  example  of  Art.  262,  the  statement 


Art.  263]  MATHEMATICAL   INDUCTION  435 

whose  general  validity  is  to  be  proved  is  suggested  in  the  first  place  by 
inspection  of  what  actually  happens  in  the  first /ew  cases.  It  is  for  this 
reason  that  the  word  induction  is  used  in  this  connection. 

The  second  part  of  the  proof  consists  in  proving  the  lemma  : 
if  the  theorem  is  true  in  the  kth  case,  it  will  also  be  true  in  the 
(^k  +  l^fh  case. 

This  usually  constitutes  the  more  difficult  part  of  the  proof.  If  we 
have  made  the  verification  and  proved  the  lemma,  tiie  proof  of  the 
theorem  is  complete.  For  the  lemma  enables  us  to  conclude,  from  the 
fact  that  the  theorem  is  true  in  case  one,  that  it  is  true  in  case  two. 
From  this  the  lemma  enables  us  to  assert  the  truth  of  the  theorem  in 
case  three,  and  so  on  for  all  cases.  The  lemma  may  or  may  not  be  true, 
but  until  we  prove  it  to  be  true  we  have  not  proved  the  general  theorem. 

In  a  proof  by  mathematical  induction  both  parts  of  the  proof 
are  equally  important. 

The  following  example,  which  is  of  considerable  interest  historically, 
will  illustrate  the  fact  that  the  first  part  of  the  proof  alone  does  not 
suffice  to  jjrove  the  theorem.     The  numbers 

2  +  1  =  3,   2-^  +  1  =  5,   24+  1  =  17,    28  +  1  =  257,   2i6  +  1  =  65537 

are  all  prime  numbers  (See  Art.  3)  and  they  are  all  expressible  in  the 
form  .,.^fc  ^  J 

Thus  the  numbers  of  the  form  2-*  +  1  are  certainly  prime  numbers  for 
k  =  0,  1,  2,  3,  4.  An  incomplete  induction  would  therefore  make  it  seem 
likely  that  all  numbers  of  this  form  are  prime  numbers.  In  fact,  Fermat 
(1601-1665)  thought  that  this  was  the  case.  But  Euler  showed  later 
(in  1732)  that  this  was  not  so  by  proving  that  the  number  given  by 
2-^  +  1  =  232  +  1  =  4,294,967,297  is  divisible  by  641. 

Another  illustration  of  this  sort,  due  to  Euler,  is  given  by  the  expression 
n'i—n  +  41.  If  we  compute  the  values  of  this  function  for  n  =  0,  1,  2, 
3,  ...,  40,  we  find  that  all  of  the  resulting  values  are  prime  numbers.  The 
evidence  seems  to  be  very  strong  in  favor  of  the  assertion  that  ifl  —  n  +  ^\ 
is  always  a  prime  number.  But  it  is  very  easy  to  see  that  this  is  not  so. 
For  n  =  41  we  find  n2  _  „  +  41  =  412,  and  this  is  obviously  not  a  prime 
number.  Two  similar  expressions  due  to  Legendre  (1752-1833)  are 
,j-2  +  n  +  17,  which  represents  prime  numbers  for  all  integral  values  of  n 
less  than  17,  and  2/t-  +  28,  which  represents  prime  numbers  for  all  values 
of  n  less  than  28. 

All  of  the  examples  given  so  far  emphasize  the  importance  of   the 


436  SEQUENCES    AND   SERIES  [Art.  263 

second  part  of  the  proof  by  mathematical  induction.  We  shall  now  give 
an  example  to  illustrate  the  fact  that  the  second  part,  when  not  accom- 
panied by  the  first  part,  does  not  constitute  a  complete  proof.  The  veri- 
fication, or  first  part,  is  also  essential. 

If  we  were  to  omit  the  verification  we  might  easily  prove  the  follow- 
ing proposition,  which  is  an  obvious  contradiction  to  the  illustrative 
example  of  Art.  262 :  the  sum  of  the  first  n  odd  integers  is  equal  to 
2.5  +  n2. 

In  fact,  the  proof  of  the  lemma  is  easy.     If 

(1)  1-^3+5+  ...  +(2k-  1)=  2.5  +  k% 
then 

(2)  1  +  3  +  5+ ...-f(2^:-l)  +  (2A:  +  l)=25  +  A:H2^  +  l=25+(A,-  +  l)2. 
From  this  it  would  follow  that 

1  +  3  +  5  +  ...  +  C2  n  -  1)  =  25  +  n^ 

in  all  cases,  if  this  equation  were  true  for  7i  —  1,  which,  however,  is  not 
the  case. 

The  process  of  mathematical  induction  may  be  compared 
to  the  process  of  climbing  up  a  ladder.  The  first  part  of 
the  proof  (preliminary  verification  for  the  first  case)  assures 
us  that  we  can  put  our  foot  on  the  ladder ;  the  ladder  actu- 
ally touches  the  ground  and  the  first  round  is  within  reach. 
The  second  part  of  the  proof  assures  us  that  there  is  no 
round  missing ;  we  can  actually  pass  from  the  ^th  to  the 
(k  +  l)th  round. 

EXERCISE  CXXIII 

Prove  the  following  formulas  and  theorems  by  mathematical  in- 
duction : 

1.  1  +  2  +  3  +  4  +  ...  +  n  =  "('^  +  ^^ . 

2.  1  •  2  +  2  .  3  +  3  .  I  +  ...  +  n(n  +  1)  =  i  n(n  +  l)(ri  +  2). 

3.  1  .2.3  +  2.3-4+    ••  +,,(«  + l)(n  +  2)=l«(n+l)(n  +  2)(n  +  3). 

4.  12  +  22  +  :',2  +  ...  +  „■!  =  1  n(n  +  l)(2n  +  1). 

5.  P  +  23  +  33  +  ...  +  „3  =  (1  +  2  +  3  +  ...  +  n)2  =  {n2(»,  -|-  1)2. 
-  1      +     1      ,    ...  +         1 


1-2      2-3  n{n  +  1)       n  +  1 

7.    2  .  4  +  4  .  (5  +  6  .  8  +  ...  +  2  /((2  /i  +  2)  =  i  «(2  n  +  2)  (2  n  +  4). 


Art. -264]  THE   SUMMATION   SIGN  437 

8.  x"  —  y"  is  divisible  by  x  —  //  if  n  is  any  positive  integer.     (See  Art. 
84.) 

9.  ./:"  +  .'/"  is  divisible  by  x  +  y  if  n  is  any  odd  integer. 
10.    .(•"  —  //"  is  divisible  hy  x  +  y  ii  n  is  any  even  integer. 

264.    The  summation  sign.     When  dealing  with  a  series 

(1)  u^  +  u^+  •••  +  M„, 

whose  terms  are  formed  according  to  some  complicated  law, 
it  becomes  very  burdensome  to  write  out  all,  or  even  many 
of  the  terms.  If  %  is  the  ^th  term,  we  may  replace  (1)  by 
the  symbol 

(2)  X  ^^^' 

which  is  read  sum  of  such  terms  as  %  from  k  =  1  to  k  =  n. 
The  2  which  appears  in  this  symbol  is  the  Greek  capital  S 
and  is  called  sigma. 

Thns  we  may  write 

n 

1  .  o  +  2  .  ;5  +  ...  +  ^•(^■  +  1)  +  -  +  n{n  +  1)  =  ^J  ^'^^  +  1)' 

*=i 

n 

1  .  2  .  3  +  2  .  :5  .  4  +  -  +  n(»  4-  1)(h  +  2)  =  J  /'(/••  +  l)^-  +  2), 


1  ,,^,^,...,^.X- 


1+12         1   +  02         1   +  :}2  1   +  ,j2         ^  1   +  ^■2 

EXERCISE  CXXIV 

1.    Use  the  summation  sign   to  represent   each   of   the  series  which 
apjiears  in  Examples  1-7  of  Exercise  CXXIII. 


Write  out.  the  first  four  tiMins  of  cacli  of  the  following  series: 
^1  +  ^-2  A   1  +  3fc  ^  /•! 


3. 


yi.  5.  yl.  ■  7.  y    '"-' 

k=\  k=\  k=\    ^  ' 


438  SEQUENCES   AND   SERIES  [Art.  265 

265.  Summation  of  a  series  whose  /tth  term  is  an  integral 
rational  function  of  k.  We  kno^v  from  the  theory  of  arith- 
metic progressions  that 

(1)  2*=^^^^. 

ft=i  -^ 

and  we  have  seen  in  Exercise  CXXIII,  Examples  2  and  3, 
that 

(2)  yt(j  +  i)  =  'i£^+ll(!i+2i, 

(3)  ^yfc(/^  +  i)(^  +  2)=<^  +  ^><^^+^><^^'+^). 

These  results  suggest  that  the  following  formula  may  be 
true 

(4)  ^^(1-Hl)...(^+^_1) 
/t=i 

_n{n->rl).  '  -{n  +  l—l^jn+l) 

l  +  l 

where  I  is  any  positive  integer,  and  where  every  term  of  the 
sum  in  the  left  member  contains  I  factors,  while  the  nu- 
merator of  the  right  member  is  a  product  of  /  + 1  factors. 
Formula  (4)  is  certainly  true  for  Z  =  1,  2,  3,  for  in  these 
cases  it  reduces  to  equations  (1),  (2),  (3)  respectively. 

To  prove  the  validity  of  (4)  in  general,  we  proceed  as  follovps.  The 
^th  terra  of  the  series  vrhich  occurs  in  (4)  is 

(5)  n,  =  ^-(^^  +  l)(^-  +  2)-(^+/-l). 

It  is  a  product  of  I  factors,  and  may  be  expi'essed  simply  in  terms  of  two 
similar  expressions,  each  of  which  contains  I  +  1  factors.  In  fact,  if  we 
put 

(6)  Vk  =  k(k  +l)(Ic  +  2)...(k  +  l-  IXk  +  0, 
and  therefore 

(7)  r,_,  =(k  -  l)k(k  +  1)  ...  (k  +  l-  2) (A-  +  1-1), 

we  observe  that  the  first  /  factors  of  r/,  are  the  same  as  the  last  /  factors 
of  J'i-i,  so  that 

Vk  -  i'A-i  =  K''  +  1)(^-  +  2)...(k  +  l-l)lk  +  l-  (k  -  1)], 


Art.  265]      SUMMATION   OF   A   CLASS  OF   SERIES  439 

or, 

(8)  V,  -  v,.t  =  k(k  +  l)(^•  +  2)  ...  (k  +  l-  !)(/  +  1). 

But  the  product  of  the  first  /  factors  of  the  right  member  of  (8)  is  equal 
to  «fc,  so  that  we  find 

(9)  v^-r,_i=(!+l)H,. 

Let  us  write  down  the  particular  equations  which  follow  from  (9)  if 
we  put  in  order  ^  =  2,  3,  4,  .•■  n.     We  find 

^'2  -  '"1        =(''  +  1)«2- 

^IQ.  '-4  -'-3  =(l+l)>^. 

?;„_!- i'„_2=^(/  + 1)«„_„      . 
r„-r„_i     =(/+!)«„. 

The  sum  of  all  of  the  left  members  reduces  to  r„  — I'l,  since  each  of  the 
other  terms  i'2.  is  •■•  i'„_i  occurs  twice,  once  with  a  plus  and  once  with  a 
minus  sign.     Therefore  we  find,  by  addition, 

(11)  I'u  -  '-i  =(/  +   l)('/2  +  "3  +  »4  +   •••   +  "„)• 

Moreover  we  have,  acccording  to  (6), 

i-i  =  1.  2.  :3.- /(/  +  !), 
and  according  to  (5) 

ui  =  1  -2  •:}.••/, 
so  that 

(12)  a=(/  +  l)"i. 

If  we  substitute  this  value  of  I'l  in  (11)  and  transpose  it,  we  find 

(•„  =(/  +  l)(Ui  +  Mo  +  ...  +  u„), 

whence 

.-..^^  ,  ,  ,  v„         n(n  +  l)(n +  2)—  (n  +1) 

(10)  III  +  Uo  +  •••   +  U„  —  ^!—  =  —5^ ^-^ ■ i- '> <-, 

^    ^  l+\  l+\ 

which  is  merely  another  way  of  writing  the  equation  (4)  which  was  to 
to  be  proved. 

Formula  (4)  is  important,  because  it  enables  us  to  find  the 
sum  of  n  terms  of  any  series  whose  kth  term  is  an  integral 
rational  function  of  k,  the  degree  of  this  function  and  its 
coefficients  being  the  same  for  all  values  of  k. 

For,  let  the  A^th  term  of  a  series  be  given  by  an  expression  of  the  form 

(14)  ut  =  ak-^  +  hk'-^  +  •..  +  mk  +  n, 

where  a,  h,  ■••  nj,  n,  and  I  are  independent  of  k.     By  the  method  of  unde- 
termined coefficients  we  can  rewrite  u^  in  the  form 


440  SEQUENCES   AND    SERIES  [Art.  265 

(15)  ui,  =  Al-(k-  +  1)  ••■  (/.•  +  /-!)+  Bk(k  +  1)  ■■■  (1-  +  I  -  2) 

+  -  +  Mk+  N, 

so  that  the  sum  ui  +  m2  •  •  •  +  M,i  will  be  equal  to 

,g.      ,  «(n  +  l)  •••  (n  +  I)       ^n(n  +  1)  •••  (»  +  Z  -  1) 
^     ^    '  /  +  1  / 

Illustrative  Example.     Find  the    sum  of   n   terms  of   the   series 
whose  ^th  term  is 

(17)  Mt  =  3A--!-4A-  +  2. 

Solution.     We  wish  to  write  iik  in  tlie  form 

(18)  w,  =  Ak{k  +\)  +  Bk  +  C. 

If  we  expand  the  products  indicated  in  (18)  we  find 

(19)  Uk  =  A  (t^  +  k)  +  Bk  +  C  =  Ak-^  +  (A  +  B)k  +  C. 

In  order  that  this  expression  may  be  identical  with  (17)  for  all  values 

of  k,  we  must  have 

A  =Z,A  +  B  =  -^,  C  =  2, 

whence 

A  =%,B  =  -'t,C  =  2, 

so  that  we  have  found 

tik  =  -ik{k^  I)  ~1  k  +  2. 

Consequently 

n  n  V 

Y^u,=  -6"2^k{k+l)-l^k  +  2n, 

k=\  fc=l  A=l 

or,  if  we  substitute  for  the  sums  which  appear  in  the  right  member  their 
values, 

(20)  2  C^'  ^■'  -  *  '^^  +  -)  =  3"<^"  +  V^"  +  ->  -  7^<^"  +  ^^  +  2  n. 


EXERCISE  CXXV 

By  til!'  method  of  Art.  26.")  prove  the  following  formulas : 

1-    V  X-2  =  y  n{n  +  1)(2  //  +  ]).  2.    y  F  =  {.'  n{n  +  \)f. 

3.    y  2  k(2  A:  +  2)  =  \  n{2  n  +  2) (2  n  +  4). 


Art.  2G6]     SUMMATIOX    OF   OTHER    SIMPLE   SERIES  441 

266.  Summation  of  some  other  simple  series.  We  may 
find  the  sum  of  n  tei-ins  of  a  series,  wliose  ^th  term  is  the 
reciprocal  of  the  ^th  term  of  series  (4)  of  Art.  265,  by  a 
very  similar  method. 

Let 

(1)  Uk=  ^ , 

kxk  +  i)(A  +  -_>)  -  (^■  +  /-l) 

and  let  us  put 

(2)  '•*  = ^ , 

-l(/--t-l)(/.-fi)  -  {k  +  l--2y 

the  inuuber  of  factors  in   the  denoniinator  of  '•/..  being  one  less  than  in 
t/*.     From  ("2)  we  find 

1 

'■'•+' ~ {k  +  l)(^•  +  -1)  :■  {k  +  i-2){k  +  i-\y 

and  therefore 


I'k  -  '•<+!  = 


—1 ri- 

{k  +  \){k  +  2) ...  (^•  +  /  -  L>)  L^-    k  +  I 

1  A-  4-  Z  -  1  -  /; 


'-] 


(k  +  1)  (^•  +  -2)  ...  (k  +  l-  2)  k(k  +  1-1) 

^ l-l 

k(k  +  l)(k  +  2)  ...  (k  +  l-  2)(k  +  /  -  1)' 
whence,  making  use  of  (1), 

(■i)  Vk  -  Vk+\  =  (I  -  l)«f 

Thus,  we  have  in  particular 

'•i  -  "2  =  G  -  l)"i. 
f2  -  rg  =  (/  -  1)^2, 

(4)  

'•u-l  -   '■„  =   ('  -   l)«n-l, 
'"n  -   ''h  +  I  =  G  —   !)««• 

From  the.se  ecjuatious  we  find  by  addition, 

'•i  -  'n+i  =  (I  -  l)("i  +  11.2  +  ■■■  +  n„), 


2"*  =  ^(''i-  '•,.+!).  if/ 9^1. 


If  finally  we  substitute  for  v^  and  (•„+!  their  values  from  (2),  we  find 

(5)     y ^ 

=-J-r-^^ 1 . 1 

;_  lL(/-  1)!      (n  +  \)(n  +2)  ••.  („  +  /  _  1)J  ' 
a  formula  which  is  valid  for  all  positive  integral  values  of  t  except  /  =  1. 


442  SEQUENCES  AND  SERIES  [Art.  266 

In  particular  for  I  =  2,  3,  4,  we  find 

V      1      -1 L_ 

^^^  ^  k(k  +  l)ik  +  2)  "  2  12!  ~  (n  +  l)(n  +  2) J  ' 

•  V 1 =iri-. 1 1. 

n  k(k  +  l)(k  +  2)(^-  +  3)      3  L  8  !      (n  +  l)(n  +  2)(n  +  3)J 

These   formulas,  and   the   method  used   in  deducing  them,  may  be 
applied  to  the  summation  of  many  other  series. 


CHAPTER   XV 

LIMITS 

267.   Limits  suggested  by  series.     Let  us  consider   again 

the  geometrical  progression 

m  1  1  1  1  ...  -i-  ... 

^^)  '  2'  4'  8'        2"-!' 

and  let  us  denote  by  S^  the  sum  of  its  first  k  terms.     Then 

S!   —  14.1—  3_9_1 
AJ2  —  ^'2  —  2  —  2' 

^3=l  +  2^+i  =  l  =  2-i, 

(2)  ^^=i  +  |+i  +  i=JJi  =  2-i, 


,V— I4-I4-I4-...4-    ^    =2 — 

and  we  observe  that  *S'„  approaches  the  limit  2  as  w  grows 
larger  and  larger.  Figure  79  illustrates  the  same  situation 
graphically.  s.   s,s,s, 

The  notion  of  limits  also  presented  itself  to    0  1  2 

our  attention  in  Arts.   87  and  89,  when  we  ^^^'  '^^ 

were  concerned  with  the  determination  of  the  tangent  at  a 
given  point  of  a  curve.  The  theory  of  limits  is  of  importance 
also  in  discussing  the  so-called  incommensurable  cases  of 
elementary  geoyietry  and  in  the  closely  allied  questions  con- 
cerning irrational  numbers.  It  also  comes  up  in  the  men- 
suration of  the  circle.  There  are  many  other  subjects  in 
pure  and  applied  mathematics  in  which  the  notion  of  limits 
is  indispensable.  The  theory  of  infinite  series,  of  which  the 
geometric  progression  (1)  is  an  illustration,  is  one  of  these. 
Since  we  wish  to  give  a  brief  treatment  of  infinite  series  in 
this  book,  we  must  first  discuss  some  of  the  more  important 
questions  which  are  connected  with  the  notion  of  a  limit. 

443 


444  LIMITS  ^  .   [AuT.  268 

268.  Definition  of  a  limit.  The  notion  of  a  variable  which 
approaches  a  limit  is  a  fairly  familiar  one.  Thus  the  sum 
*S'„  of  n  terms  of  the  geometrical  progression  (1),  Art.  267, 
approaches  the  limit  2.  If  P„  denotes  the  perimeter  of  a 
regular  polygon  of  n  sides  circumscribed  about  a  circle 
whose  circumference  is  (7,  P„  approaches  Q  as  a  limit.  At 
the  same  time  the  area  A^  of  the  polygon  approaches  the 
area  A  of  the  circle  as  a  limit,  and  the  same  statements  are 
true  of  the  perimeters  and  areas  of  the  regular  inscribed 
polygons.  When  an  automobile  slows  up  and  finally  stops, 
it  approaches  its  stopping  place  as  a  limit.  If  we  prefer  to 
think  of  the  variable  as  a  number^  we  may,  in  this  last'illus- 
tration,  say  that  the  distance  from  the  starting  place  of  the 
automobile  approaches  as  a  limit  the  distance  from  that  point 
to  the  place  where  it  stops.  A  pendulum  which  swings  in  a 
resisting  medium  like  air  approaches  the  vertical  position  as 
a  limit.  In  Arts.  87  and  89  we  developed  the  notion  that  a 
straight  line  which  joins  a  fixed  point  of  a  curve  to  a  mov- 
ing point  of  the  same  curve,  approaches  a  limiting  position, 
called  a  tangent  of  the  curve,  when  the  moving  point  ap- 
proaches the  fixed  point  as  a  limit. 

In  order  that  we  may  be  able  to  reason  logically  about 
limits  in  general,  we  must  think  about  these  various 
instances,  discover  the  essential  feature  which  they  all  have 
in  common,  and  then  formulate  a  definition  which  shall 
cover  them  all.  If  we  do  this  we  are  led  to  the  following 
definition. 

A  variable  x  is  said  to  approach  the  constant  a  as  a.  limits  if 
the  law  ivhich  describes  the  variation  of  x  is  such,  that  the 
numerical  value  of  the  difference  between  a  and  x  ivill  ultimately 
become  and  remain  smaller  than  any  positive  number  which  may 
have  been  assigned  in  advance. 

B}^  the  numerical  value  of  the  difference  between  a  and  x 
we  mean,  as  usual,  the  magnitude  of  this  difference,  no  atten- 
tion being  given  to  its  si  n.  We  shall  denote  the  numerical 
value  of  this  difference  by  the  symbol  \x  —  a|,  as  in  Art.  17. 


Art.  268]  DEFINITIOX   OF   A   LIMIT  445 

The  positive  number  assigned  in  advance,  which  is  men- 
tioned in  the  definition,  may  be  called  8.  If  we  use  these 
notations,  we  may  reformulate  our  definition  as  follows: 

A  variable  x  is  said  to  approach  the  constant  a  as  a  limit,  if 
the  law  according  to  ivhidi  the.  variable  x  changes  is  such  as  to 
insure  that  uldmately  \x  —  a\  will  become  and  remain  less  than 
ang  positive  number  h  which  has  been  previously  selected,  and 
which  mag  be  chosen  as  S)nall  as  we  please. 

Thus,  to  test  whether  a  variable  x  approaches  a  certain 
number  a  as  a  limit,  we  may  proceed  as  follows  : 

Step  1.  Choose  a  positive  number  8.  It  is  understood 
that  this  number  may,  in  particular,  be  chosen  arbitrarily 
small. 

Step  2,  Examine  whether  the  law  according  to  which 
the  variation  of  x  takes  place  will  permit  |a:  —  a|  to  become 
less  than  8. 

Step  3.  Examine  the  variation  of  x  after  this  stage  has 
been  reached,  to  see  whether  ja;  —  a|  will  not  merely  become, 
but  ever  afterward  remain,  less  than  8. 

Step  3  is  very  essential.  Thus  in  the  geometric  progression  1,  ^,  \,  \,  ••• 
of  Art.  2(57,  the  numerical  value  of  S„  —  1-5/8  does  become  less  than  any 
positive  number  8  for  n  =  4.  In  fact  we  have  St  —  15/8  =  0.  But  S^ 
does  not  approach  15/8  as  a  limit,  since  j  S,,  —  15/8  |  does  not  remain 
less  than  8  after  n  has  increased  beyond  the  value  n  =  4. 

In  accordance  with  our  definition  x  may  approach  the  limit 
a  from  above,  so  that  x  —  a  is  always  positive,  or  from  below, 
so  that  X  —  ah  always  negative.  Or  else  x  —  a  may  be  posi- 
tive during  some  of  the  stages  of  the  approach  and  negative 
during  others.  Since  the  definition  merely  speaks  about 
the  numerical  value  of  x—a,  all  of  these  cases  are  equally 
admissible. 

The  geometric  progression  of  Art.  267  is  an  instance  of  a  variable 
which  approaches  its  limits  from  below.  The  area  ^„  of  a  regular  cir- 
cum.scribed  polygon  aj^proaclies  the  area  of  the  circle  from  above.  If  we 
consider  a  sequence  of  regular  polygons  alternately  inscribed  and  cir- 
cumscribed about  a  circle,  the  area  of  a  polygon  of  this  sequence  also 


446  LIMITS  [Art.  268 

approaches  the  circle,  but  alternately  from  below  and  above.  The  same 
thing  is  trne  of  a  geometric  progression  whose  common  ratio  is  negative 
but  numerically  less  than  unity. 

Again,  ovir  definition  says  nothing  about  whether  the  variable 
reaches  its  limit  or  whether  it  does  not.  The  variable  S^  of 
Art.  267,  ,  . 


'S'n  =  l+f,+    ••■    + 


—  9   — 


1 


9n-l  On-1 


never  reaches  its  limit  2  for  any  value  of  n.  Neither  do  the 
variables  P„  or  A^-,  if  P„  and  J.„  denote  the  perimeter  and 
area  of  the  regular  circumscribed  n-gon.  But  an  automobile 
not  only  approaches  its  stopping  point  as  a  limit,  but  reaches 
it.  Therefore,  the  variable  distance  from  a  fixed  starting 
point  to  an  automobile  approaches  a7id  reaches  a  limit  when 
the  automobile  stops. 

Thus,  a  variable  which  approaches  a  limit,  may  or  may  not 
reach  its  limit.* 

It  is  true,  however,  that  in  many  problems  it  is  desirable, 
or  even  necessary,  to  think  of  the  variable  x  as  approaching 
its  limit  a  without  reaching  it.  But  such  a  restriction,  if 
desired,  we  shall  add  explicitly.  It  is  not  included  auto- 
matically when  we  say  that  x  approaches  a  as  a  limit. 

We  use  the  symbols 

lim  X  =  a         (read  the  limit  of  x  is  eqnal  to  a), 
or  X — ^a         (read  x  approaches  a  as  a  litnit^, 

whenever  we  ivish  to  say  that  a  variable  x  approaches  a  constant 
a  as  a  limit. 

The  definition  of  a  limit  leads  at  once  to  the  following 
property,  which  is  used  very  frequently. 

If  trvo  variables.,  x  and  y.,  simultaneously  pass  through  the 
same  values,  so  that  x  and  y  are  equal  to  each  other  at  each  and 

*  This  remark  is  valid  on  the  basis  of  the  definition  of  limit  as  we  have  formu- 
lated it  in  tliis  bool^.  Some  authors  define  a  limit  differently,  including  in  their 
definition  explicitly  a  statement  that  the  variable  shall  not  reach  its  limit.  For 
some  purposes  this  definition  is  preferable  to  ours.  But  it  leads  to  some  compli- 
cations which  we  prefer  to  avoid. 


Art.  269]  IXFINITY  447 

every  stage  of  their  variation,  then  if  one  of  these  variables 
approaches  a  limit,  so  does  the  other,  and  their  limits  are  equal. 
In  symbols;  if  x  =  y,  and  if  liin  .v  =  «,  theyi  Vww  y  =  a. 

269.  Infinity.  The  simplest  sequence  of  numbers  is  that 
of  tlie  positive  integers 

(1)  1,  2,  3,  ...,  k,  :.. 

According  to  our  fundamental  assumptions  (see  Arts.  1  and 
2),  if  k  is  a  positive  integer,  no  matter  how  large,  there 
always  exists  another  one  still  larger,  namely  ^  +  1.  Con- 
sequently the  sequence  (1)  has  no  last  or  largest  number; 
it  has  no  limit,  it  is  unbounded.  A  variable  which  assumes 
such  values  is  said  to  grow  beyond  bound,  or  to  become 
infinite. 

j\Iore  generally,  if  the  law,  according  to  ivhich  a  variable  x 
changes,  is  such,  that  the  numerical  values  of  x  ultimately  he- 
come  and  remain  greater  than  any  positive  number  M  chosen  in 
advance,  x  is  said  to  become  infinite.  This  is  often  expressed 
by  the  symbols 

(2)  lima:=x     or   x — >-xi. 

These  symbolic  statements  are  in  common  use,  and  we,  therefore,  find 
it  necessary  to  become  acquainted  with  tliem.  It  should  be  remembered, 
however,  that,  strictly  speaking,  the  use  of  the  symbol  lim  in  this  case  is 
inappropriate.  For  when  we  write  (2)  we  mean  to  indicate  that  x  does 
not  approach  a  limit,  but  grows  numerically  beyond  bound.  Tu  fact,  the 
word  infinite  means  just  this;  unbounded  or  unlimited. 

In  accordance  with  this  dcluiition  a  variable  x  may  become 
infinite  by  passing  through  a  sequence  of  values  such  as  (1), 
all  of  which  are  positive,  or  by  passing  through  a  sequence 
of  values  all  of  which  are  negative,  or  else  by  passing 
through  values  some  of  which  are  positive  and  some  of 
which  are  negative.  If  we  restrict  the  variation  of  x,  by 
compelling  x  to  assume  only  positive  values  while  it  is  grow- 
ing beyond  bound,  we  write 

lim  X  =  -|-  GO. 


448    .  LIMITS  [Arts.  270,  271 

Similarly,  the  symbolic  equation 

lim  X  =  —  CO 

means  that  x  grows  beyond  bound,  exclusively  through  nega- 
tive values.      This  distinction  is  frequently  very  important. 

270.  Infinitesimals.  If  a  variable  approaches  zero  for  its 
limit,  if.  is  called  an  infinitesimal. 

The  following  statements  are  immediate  consequences  of 
this  definition. 

1.  If  X  approaches  the  limit  a  then  x  —  a  is  an  infinitesimal. 

2.  If  X  becomes  infinite,  the  reciprocal  of  x  is  an  infinitesimal. 

3.  Ifx  is  an  infinitesimal,  the  reciprocal  of  x  becomes  infinite. 

The  last  two  statements  are    sometimes  expressed   symbolically  by 

writing  j  ^ 

—  =  0,    -  =  CO. 
CO  0 

Literally  these  equations  have  no  iiieauing,  since  division  by  zero  is  ex- 
cluded from  algebra  (Art.  21)  and  since  the  symbol  -jo  does  not  stand  for 
a  number. 

271.  Variables  which  remain  finite.  If  a  variable  x  be- 
comes infinite  it  has  no  limit  although  we  write  symbolically 
lim  X  =  cc  .  But  a  variable  may  remain  finite  and  neverthe- 
less not  approach  a  limit.  Thus,  if  x  assumes  in  succession 
the  values  +1,  —  1,  -f  1,  —  1,  and  so  on,  it  remains  finite 
and  nevertheless  it  does  not  approach  a  limit. 

A  variable  x,  whether  it  approaches  a  limit  or  not,  is  s<tid  to 
remain  finite,  if  there  exists  a  positive  constant  M,  such  that 

\x\  <  M 

for  all  of  those  values  which  x  may  assume  in  accordance  /rifh 
the  law  vhi'di  regulates  its  variation.  Such  variables  are  also 
said  to  be  bounded. 

In  particular,  a  variable  which  approaches  a  limit  may  be 
regarded  as  belonging  to  this  class  of  variables  which  remain 
finite.  (See  Art.  268.)  An  extremely  special  case  of  such 
a  variable  is  a  constant. 


Art.  27-2]     A    THEOREM    ABOUT   INFINITESIMALS  449 

272.    A  theorem  about  infinitesimals.     The  following  theo- 
rem will  tind  frequent  application  in  what  follows. 

Let  u  and  V  he  two  infinitesimals,  so  that 

(1)  lim  u=  0,    lini  r  =  0, 

and  let  X  and  Y  he  two  variahles  which  remain  finite.  Then 
Xu  +  Yc  trill  he  an  infinitesimal,  that  is, 

(2)  lim  (Xu  +  Yv)  =  0. 

Proof.     Since  X  and  Y  remain  tinite,  there  exists  a  finite 
positive  number  M,  such  that 

(3)  \X\<M,^Y<M.     (See  Art.  271.) 

Since  u  and  v  are  infinitesimals,  we  may  choose  a  positive 
number  8,  as  small  as  we  please,  and  then  be  sure  that  the 
values  assumed  by  u  and  v  will  ultimately  become  and  re- 
main so  small  that 

(4)  I  M  I  <  8,    1 1;  I  <  8.      (See  Art.  270.) 

As  a  consequence  of  (3)  and  (4)  we  shall  have 

I  Xu  \<MS,    \Yv\<  MS, 
and  therefore 

(5)  \XuA-Yv\<\Xu\  +  \Yv\<2  MS.     (See  Art.  22.) 

But  the  right  member  of  (5)  may  be  made  smaller  than  any 
positive  number  S',  no  matter  how  small.  To  show  this,  we 
think  of  S'  as  being  assigned  first,  as  small  as  we  please. 
Then,  in  the  second  place,  we  choose  S,  so  that 

'<2M- 
Then  we  shall  have 

(6)  2MS<S'. 

Thus,  if  u  and  v  are  infinitesimals  and  if  X  and  Y  remain 
finite,  the  inequalities  (5)  and  (6)  assure  us  that  ultimately 


460  LIMITS  [Art.  273 

I  Xu  +  Yv  I  will  become  and  remain  less  than  h\  where  S'  is 
an  arbitrarily  small  positive  number.  But  this  means  that 
Xu  +  Yv  approaches  the  limit  zero  (see  Art.  268),  so  that 
Xu+  Yv  is  an  infinitesimal.      (Art,  270.) 

This  theorem  may  obviously  be  extended  to  any  sum  com- 
posed of  a  finite  number  of  terms,  as  follows. 

If  Ui^  u^-,  •••,  Un  <^'*^  a  finite  number  of  infinitesiinals,  and  if  X^^ 
X^i  •••  X,i  are  n  variahles  which  remain  finite,  then  X-^u^-^-  X^u^ 
+  •••  +  X,^nn  is  an  infinitesimal. 

273.  Theorems  about  limits.  Let  x  and  y  he  tivo  variables, 
each  of  which  approaches  a  limit,  and  let 

lim  X  =  a,    lim  y  =  b. 

Then  ive  shall  have 

(1)  lim  (.r  +  ?/)  =  a  +  ^  =  lim  x  +  lim  y, 

(2)  lim  (x  —  y^=  a  —  h  =  lim  x  —  lim  y, 

(3)  lim  xy  =  ab  =  (lim  a:) (lim  ?/). 

Proof.     To  prove  (1)  we  put 
(4)  u  —  X  —  a,    V  =  y  —  b. 

Then  u  and  v  are  infinitesimals.  (Theorem  1,  Art.  270.)  We  now 
apply  the  theorem  of  Art.  272  putting  A' =  Y=  1.  We  conclude  that 
u  +  V  will  be  an  infinitesimal,  that  is, 

lim  [.r  -  «  +  ?/  -  i]  =  lim  [./:  +  //  -  (a  +  b)']  =  0, 

proving  that  x  +  y  ajijiroaches  a  +  &  as  a  limit. 

To  prove  (2)  we  use  the  conclusion  from  Art.  272  that  u  ~  v  will  be 
an  infinitesimal. 

To  prove  (3)  we  write  (4)  as  follows 

X  =  a  +  u,  y  =  h  +v, 
whence,  xy  =  ab  +  bu  +  av  +  uv  =  ab  +  [bu  +  (a  +  u)v'\, 

or  xy  —  ab  =  bu  +  (a  +  ii)i\ 

We  may  now  apply  the  theorem  of  Art.  272,  putting 
A'  =  b,    Y  =  a  +  w, 

since  ti  and  v  are  infinitesimals,  so  that  A'  and  Y  remain  finite.  We  con- 
clude that  xy  —  ab  is  an  infinitesimal,  and  therefore 

lim  xy  =  ab  =  (lim  a;) (lim  y). 


Art.  274]  LIMIT   OF   A   QUOTIENT  461 

Thus  we  have  proved  the  following  theorem.  If  each  of 
two  variables  approaches  a  limit,  so  does  their  sum,  their  differ- 
ence, and  their  product.  Moreover  the  limit  of  the  snm  of  the 
variables  is  equal  to  the  sum  of  their  separate  limits,  and  a 
similar  statement  holds  for  the  limit  of  the  difference  and 
the  limit  of  the  p)roduct. 

It  is  obvious  that  this  theorem  may  be  extended  to  sums, 
differences,  and  products,  composed  of  any  finite  number  of 
terms  or  factors.  From  this  remark  we  obtain  the  following 
simple  corollaries. 

Cor.   1.     If  X  approaches  a  as  a  limit,  and  if  n  is  a  positive 

integer^  then 

lim  j;"  =  a"  =  (lim  a:)". 

Cor.  2.     If  c  is  any  constant.^ 

lim  ex  =:  ca  =  c  lim  x. 

By  combining  these  theorems  in  obvious  fashion,  we  find 

Cor.  3.  Let  f(x)  be  any  integral  rational  function  of  x, 
and  let  a  be  any  finite  number.  If  x  varies  in  such  a  way  as 
to  approach  a  as  a  limit,  then  the  corresponding  values  of  f{x) 
will  also  approach  a  limit,  and  moreover 

lim/(  a;)  =/(«). 

274.   Limit  of  the  quotient  of  two  variables.     Let  x  and  y 

be  two  variables  which  approach  the  limits  a  and  b  respect- 
ively.    Then  we  may  write 

X  =  a  +  u,    y  =  b  -\-v, 

where  u  and  v  are  infinitesimals,  and 

X  _a  -\-u 
y      b  -\-v 
Therefore 

m  ^-'L  =  'L±JL^^=^''—£^=Xu+Yv, 

^  ^  y      b      b  +  v      b      bib  +  v^} 


452  LIMITS  [Art.  274 

if  we  put 

C2\  X  =  Y  =      ~  ^ 

If  b  is  not  equal  to  zero,  X  and  1^  approach  the  limits  1/b 
tind  —  a/b"^  respectively,  and  therefore  remain  finite.  (Art. 
271.)  Consequently  (See  Art.  272)  the  right  member  of 
(1)  is  an  infinitesimal,  and  we  conclude 

(3)  Inn  -  =  -  = it  ^  ^  0. 

7/      b      lim  ?/ 

This  conclusion  does  not  follow,  by  our  method  of  proof, 
if  5  =  0.  For  in  that  case  the  variables  X  and  Y,  defined  by 
(2),  do  iiot  remain  finite.  But  we  may  say  more  than  that. 
The  theorem  certainly  is  not  true  when  6  =  0.  For,  in  that 
case,  lira  y  =  0,  and  equation  (3)  cannot  be  true,  since  the 
riglit  member  of  (3)  becomes  meaningless  when  5  =  0. 

We  may  express  our  result  as  follows: 

Theorem  1.  If  each  of  two  variables  approaches  a  limit, 
their  quotient  also  approaches  a  limit,  which  is  equal  to  the 
limit  of  the  dividend  divided  by  the  limit  of  the  divisor, 
provided  that  the  latter  limit  is  different  from  zero. 

This  theorem  and  the  above  discussion  should  not  be 
understood  to  imply  that  the  quotient  x/y  may  not  have  a 
limit,  even  if  the  limit  of  y  is  equal  to  zero.  We  have  only 
asserted  that  this  limit,  if  it  exists,  is  not  equal  to 

lim  X 
lim  y 

since  this  last  expression  is  meaningless  when  lim  y  =  0. 

Theokem  2.  If  the  limit  oj'  y  is  equal  to  zero,  the  quotient 
x/y  may  or  may  not  have  a  limit.  The  decision  of  this  ques- 
tion, in  any  particular  instance,  depends  upon  the  relation  (if 
any')  whi'-h  exists  between  the  laws  accordiny  to  which  the 
variables  x  and  y  pass  through  their  various  values. 

We  certaiidy  cannot  expect  x/y  to  ap})roach  a  limit  if 
lim  y  =  0,  and  if  x  does  not  approach  a  finite  limit.     Let  us 


Akt.  274]  LIMIT   OF   A   QUOTIENT  453 

then  assume  that  x  does  approach  a  finite  limit  a  which  is 
different  from  zero.     In  this  case  x/y  will  become  infinite. 

The  following  example  will  illustrate  this  statement.     Let  x  assume 
in  succession  the  following  values 

x=\^y      1+1,   l+L'-,    l+l,liinx=l, 
2  4  8  2" 

and  let  the  corresponding  values  of  y  be 

111  1      r  A 

11=         -,  -.  -,  •••  — ,  lim  M  =  0. 

■'  2  4  8  2''-  ^ 

Clearly  the  values  of  the  quotient 

■!:=         ;5,  5,  !J,..-,    2"  +  l,    ... 

U 

will  grow  beyond  bound,  that  is, 

I-         X 

lini  -  =  X). 
!J 

Biity  if  X  and  y  both  (tpproacli  the  limit  zero,  it  is  impossible 
to  say  in  general  whether  the  quotient  approaches  a  limit  or  not. 

Thus,  let  X  assume  the  values 

(1)^=7-    o»     o'   .-,•••,     limx  =  0; 

1        2         O  71 

and  let  us  consider  the  case  where  the  corresponding  values  of  //  are 

(2)  //  =  !,  -L,  -L,...     1  ,...,   mny  =  o. 

1     -^2      V:]  Vn 

Then  r///  will  assume  in  succession  the  values 

Gi)  --L   — ,   -^, .-^'•••.    li"'---0. 

//  V2     V;s  V?t  // 

If,  instead,  the  values  of  //.  wliich  (•orn'S])()nd  (o  the  values  (1)  of  x 

are  ... 

(1)  ,,  =  ■'     '1     2. ,  ■!....,       lim^  =  0. 

1      2      •)  n 

the  limit  of  .r///  ^^'i"  ''^  ('(lual  to  1/5.     Again,  if  the  values  of  //  are 

(.1)   v  =  ^,    1,     i, -,■■■,       lim//  =  0, 

1       2-      •>'-  n- 


464  LIMITS  [Art.  275 


the  sequence  of  values  for 

^hj 

\A 

'ill  be 

X  _ 

1, 

,  2,  3,   -.jn,  . 

y 

so  that,  in  this  case, 

1 

im  -  =  GO . 

y 

Finally,  if  we  should  liave 

1 

- 

1 

+  3'        4' 

±1 


the  quotient  would  assume  the  values 

^=+1,   -1,   +1,   -1,-, 

so  that  xl]i  remains  finite  but  does  not  approach  a  limit. 

Thus,  if  both  x  and  y  approach  the  limit  zero,  the  quotient 
xjy  may  approach  the  limit  zero,  it  may  become  infinite,  it 
may  approach  a  finite  limit,  or  finally  it  may  remain  finite 
without  approaching  a  limit.  To  decide  what  becomes  of  the 
quotient  x/y^  if  both  x  and  y  approach  zero  as  a  limit,  requires 
special  investigation  in  every  particular  case. 

EXERCISE  CXXVI 

Investigate  whether  the  limits  indicated  in  the  following  examples 
exist  or  not.  Find  the  limit  when  it  exists,  quoting  the  theorems  needed 
in  your  argument  at  every  step. 

1.      lim  f2  +  1 


2.     lini 

n — ^-00 


lim  r\  +  (-l)n2l. 


3. 

li  m   (3  X  - 

-  5).              5. 

\im(x+7)(x-S) 

x->4 

4. 

lim  (2  X  - 

!)•              6. 

lim2-  +  l. 
x_>i  X  +  3 

8. 

2-1 
n 

lllll 

9.     lim  5  n-, 

"-^3+- 

>i— >-« 

7. 

n    >°o 


275.  Limit  of  the  nth  power  of  a  positive  number  as  n  grows 
beyond  bound.  Let  r  be  a  positive  number  and  let  us  con- 
sider the  sequence  of  numbers 

r^,  r^,  r^,  •••,  r",  ••• 

for  all  positive  integral  values  of  w. 


Art.  270]  CONTINUITY  OF   A   FUNCTION  455 

If  r  =  1,  we  have  r"  =  1"  =  1,  and  therefore 
(1)  lim  1"  =  1. 

If  ?•  >  1  let  us  put 

r=l  +  h 

wliere  /;  is  a  positive  number.     By  the  binomial  theorem  we  have 

r"  =(1  +  h)"  =  1  +  )ih  +  positive  terms        (See  Art.  SS). 
so  that 

r"  >  1  +  7ih. 

Clearly  we  may  choose  n  so  great  that  1  +  ////  will  become  and  remain 
greater  than  any  jiositive  number  ^f,  no  matter  how  large  M  may  be. 
It  suffices  for  this  purpose  to  choose 

^^-  1 


for  then  we  shall  have 


">    /, 


1  +nJi>l  +'^^^^h  =  M. 
h 


In  other  words;  if  r  is  a  positive  number  greater  than  unity,  the  ex- 
ponent n  may  be  chosen  so  great  as  to  cause  r"  to  become  and  remain 
larger  than  any  positive  number  ^f.  That  is.  r"  becomes  infinite,  or  in 
symbols 

(2)  lim  r"  =  X   if  ?>1. 

If  ;•  <  1,  we  may  put  r  —  — ,     r'  >  1, 

and  r"  = 

We  .shall  have    lim   (;•')"  =  oo.  since  ?•'>  1.     Therefore  (see  Art.  270), 

(3)  lim  ;•"  =  0  if  r<l. 

We  may  generalize  our  result  slightly.  Let  r  he  any  real 
number^  positive  or  negative^  and  let  \r\  denote  the  numerical 
vahie  of  r.      Then  we  shall  have 

lim  |r"I  =  GO  if  |r|  >  1, 
lim|r"l  =  l  if  |r|  =  l, 
lim|r«|  =  0    if  \r\  <  1. 

276.  Continuity  of  a  function.  Let  a  and  h,  where  b  >  a, 
be  two  real  numbers.  By  the  interval  (a  •••  b)  we  mean  the 
assemblage  of  all  real  numbers  between  a  and  i,  includ- 
ing the  numbers  a  and  b  themselves.       Any  such  number, 


456 


LIMITS 


[Art.  276 


excepting  a  and  b,  is  said  to  be  m  the  interior  of  the  interval 
(a  •••  b). 

Let  JO  represent  any  number  in  the  interval  («  •••  b^.  We 
say  that  afunctio7i  of  x  is  defined  for  x=  p,  if  it  is  clear  from 
the  definition  of  tlie  function,  as  expressed  by  a  formula  or 
by  some  other  description  of  the  function,  what  tlie  value  of 
the  function  will  be  for  re  =  p.  The  function  is  delined  as 
a  real  oyie-valued,  function  for  x  =  p  if  there  is  only  one  value 
prescribed  for  x  =  p  and  if  this  one  value  is  a  real  number. 
Let  y=f{x)  be  defined  as  a  real  one- 
valued  function  of  x  for  all  values  of  x  in 
the  interval  (a  •••  6),  and  let  us  think  of 
Fig.  80  as  representing  the  graph  of  such 
rx     a  function,  where  we  have  made 


6)^  =  a,      OB  =  b,     OX=x. 

and  where  the  values  of  f(a),  /(^),  and  f(x)  are  repre- 
sented by  the  ordinates  AA',  BB\  and  XX'  respectively, 

so  that     ^^,  ^^^^^^     BB'^fib),     XX' =  fix). 

Let  lis  now  pick  out  any  particular  point  P  of  the  interval 
AB,  whose  abscissa  is  p,  so  that 

a<p<h,    fip)=FF'. 

If  the  curve  is  continuous  or  unbroken,  as  represented  in  the 
figure,  it  is  clear  that  a  variable  ordinate  XX'  will  approach 
the  fixed  ordinate  PP'  as  a  limit  if  X  approaches  P  as  a 
limit  in  any  manner,  that  is,  from  the  right,  or  from  the  left, 
continuously,  or  by  a  series  of  jumps.  In  other  words  we 
shall  have 

(1)  lini  f(x)  =  f(p). 

J-    p 

But  if  there  is  a  break  in  the  continuity  of  the  curve  at 
PP',  (see  Fig.  81)  XX'  will  approach  a  different  limit 
{PP'  or  PP"y  according  as  X  approaches  P  from  the  right 
or  from  tlie  left.  If  X  approaches  P  as  a  limit  in  an  arbi- 
trary fashion,  partly  from  the  right  and  partly  from  the  left 


*y      t  ,, 


/>■ 


A        P  X    li 
Fiu.  81 


Art.  27G]  CONTINUITY   OF    A    FUNCTION  457 

(as  is  permissible  according  to  tlie  definition  of  a  limit),  XX' 
will  oscillate  between  values  nearly  equal  to  PP'  and  PP" 
respectively,  and  not  approach  any  limit. 

We  see  that  the  possibility  of  such 
a  break  in  the  continuity  of  the  graph 
is  excluded  if  equation  (1)  is  satisfied. 
This  remark  leads  to  the  following  defini- 
tion of  continuity. 

A    real    one-valued    fnnction   f(x}    is 
said  to  be  continuous  in  the  vicinity  of  a  particular  value 
x  =  p  ii  the  following  conditions  are  satisfied. 

1.  The  function  f(x')  is  defined  for  t  =  p,  that  is,  the  defini- 
tion of  the  function  assigns  a  unique  definite  finite  value  to  the 
sijmhol  fi^p}-  This  symbol  f(^p)  then  represents  a  definite 
finite  real  number. 

2.  TJie  function  f(^x)  is  defined  for  all  values  of  x  in  the 
neighborhood  of  x  =  p,  in  such  a  tvay  that  when  x  approaches  p 
as  a  limit,  the  function  f{x)  tvill  approach  one  and  the  same 
definite  finite  number  as  a  limit,  7io  matter  according  to  what 
particular  laiv  x  may  approach  p.  This  is  expressed  by  saying 
that  the  limit  lim  f(^x)  exists. 

3.  Finally,  if  the  function  is  to  be  continuous,  the  condition 
lim  f(x)=f{p^  must  be  satisfied. 

This  last  statement  is  read  as  follows;  the  limit  of  f(.r),  as  x  aj> 
proaches  p,  is  equal  to  f(  p). 

If  the  function  f(x')  is  defined  only  over  a  finite  interval 
(a  •••  /))  we  should  modify  Part  2  of  the  above  definition  in 
the  case  p  =  a  ov  p  =  b.     We  shall  say  that  lim  f(x}  exists 

if  we  obtain  a  definite  finite  limit  for  f(x)  when  x  ap- 
proaclies  a  from  above,  since  no  other  values  of  x  would  be 
admissible  in  su(;h  a  case.      Similarly  for  lim/(.r). 

If  a  function  f  (^x')  is  continuous  in  the  vicinity  of  every  par- 
ticular value  p  which  belongs  to  the  interval  (<*•••  6),  it  is  said 
to  be  continuous  in  the  whole  interval. 


458  LIMITS  [Akt.  277 

We  have  seen  in  Art.  273  that,  if  fix)  is  an  integral 
rational  function  of  a:,  then 

lira  f<ix~)=f(p), 

if  p  is  any  finite  number.  Therefore  we  may  now  state  the 
following  theorem. 

An  integral  rational  function  of  x  is  continuous  for  all  finite 
values  of  x. 

This  is  a  theorem  which  we  have  stated  and  used  before. 
(See  Art.  96.)  But  up  to  the  present  moment  we  have  left 
it  unproved. 

277.  Continuity  of  a  fractional  rational  function.  Let  us 
consider  next  a  fractional  rational  function  (see  Art.  135), 

(1)  R(^=^^ 

where /(a;)  and  g(x')  are  integral  rational  functions  of  x. 
If  such  a  function  is  not  continuous  in  the  neighborhood  of 
a  particular  value  x  =  p,  according  to  the  definition  of  con- 
tinuity (Art.  276),  this  may  be  due  to  any  one  of  the  follow- 
ing reasons. 

1.  The  function  may  not  be  defined  for  x  =  p. 

2.  The  function  may  not  have  a  limit  when  x  approaches 
p.  This  may  happen  because  tlie  function  grows  beyond 
all  bound  when  x  approaches  p,  or  because  the  limit  is 
not  the  same  when  x  approaches  p  in  several  different 
manners. 

3.  The  limit  of  R(x')  as  x  approaches  p  may  not  be  the 
same  as  the  value  -B(/')  which  is  obtained  by  substituting 
x  =  p  in  the  defining  expression  (1)  of  R(x). 

Let  us  examine  these  possibilities.  Since  f(x)  and  g  (x') 
are  integral  rational  functions,  they  are  defined  for  all  finite 
values  of  x.  Therefore  M(x),  their  quotient,  will  be  defined 
for  all  values  of  x  except  for  those  which  cause  the  divisor 
g(x^  to  assume  the  value  zero.  (See  Art.  21.)  l^et  x  =  p 
be  such  a  value  of  x,  so  that  g(^p^=  0.     If  the  rational  func- 


Art.  277]     CONTIXriTY   OF    A    RATIONAL   FUXCTIOX       459 

tion  is  in  its  lowest  terms  (see  Art.  137)  we  shall  then  have 
/(jy)9tO,  and  x  =  p  will  be  a  pole  of  the  fractional  function 
rIx).     CSee  Art.  130.)     In  that  case 

(1)  lim  It{x)=  00  ; 

in  other  words,  IK^x}  has  no  finite  limit  as  x  approaches  p. 

Therefore,  a  fractional  rational  function  is  discontinuous  in 

the  neighborhood  of  any  one  of  its  poles. 

If  the  denominator  g{x^  is  not  equal  to  zero  for  x  =  p,  we 

have 

lim  /(a:) 

(2)  lim  i^(.r)  =  f^^^L^ (Theorem  1,  Art.  274.) 

But  since /(a;)  and  (/(^x)  are  integral  rational  functions,  and 
are,  therefore,  continuous  for  all  finite  values  oi  x  (Art.  276), 
we  have  further 

lim/(a:)  =  /(^),     lim  g(x)  =  g(p)=^0. 

x-^p  x-^p 

(Art.  276,  Definition  of  a  continuous  function.) 
Consequently  we  find  from  (2), 

lim  i2  (a:)  =  ^^^  =  i^  (  7^) . 

But  this  means,  according  to  the  definition  (Art.  276),  that 
R(x)  is  continuous  in  the  neighborhood  of  x  =  p. 

We  have  proved  the  following  theorem.  A  fractional 
rational  function  of  x  in  its  lowest  terms  is  continuous  in  the 
neighborhood  of  every  finite  value  of  x,  excepting  only  those 
which  cause  the  denominator  to  assume  the  value  zero. 

EXERCISE     CXXVII 

1.    Find  the  limit  which   —   approaches  when  x  approaches  the 

limit  4.  ^  "•"  ^ 

x8  —  2 

Siilution.     The  function      f(^)  = ~ 

a:  +  1 


460  LIMITS  [Art.  278 

is  rational,  and  is,  therefore,  a  continuous  function  in  the  neighboi'hood 
of  all  values  of  x  which  are  not  poles  of  this  function.  The  only  pole 
of  this  function  is  a:  =  —  1.  Therefore  /'(x)  is  continuous  in  the  neigh- 
borhood of  X  =  4  ;  and  consequently,  using  the  definition  (  I  a  continuous 

function 

lim/(x)  =  /(4), 

—  2      4^  —  2      62 


or  lim  ,         ,       ,        . 

j:-^  X  +  1         4+1         o 

Find  the  limits  indicated  in  the  following  examples.     Give  the  reason- 
ing as  in  Example  1. 

2.  lim — — ■■  4.     hm  " 6.     lim  — ■ 

j->0  X  —    1  x->0  X^   +  X  +   1  i->0  X 

3.  hin 5.      lim  "      ' — 


j:_>1  X-  +  X  +  1  x-^-l  x^  —  1  x->- 1  X  -1-  1 

_        , .        X^  +   1  0        1-^+7 

8.     hill 9.     hm 

i_>.l  x^  —  1  x-^\  X  —  1 

278.  Indeterminate  forms.  We  may  summarize  the  prin- 
cipal results  of  Art.  277  as  follows.  If  R{x)  is  a  rational 
function  of  x,  written  in  its  lowest  terms,  and  if  x  ajjproaches 

the  limit  p,  then  ^ 

^  lim  B(x)  =  Jl(p). 

Unless  JO  is  a  pole  of  RQx'),  this  limit  will  be  a  definite  finite 
number. 

If  R(x')  is  a  rational  function  which  is  not  in  its  lowest 
terms,  and  if  we  write  /.^  x 

then  /(a;)  and  gQx'),  the  numerator  and  denominator  of 
IK^x^,  will  have  a  common  factor  dependinq-  upon  x.  If  we 
divide  botli  numerator  and  denominator  of  R(^x^  by  tlieir 
highest  common  factor,  we  sliall  obtain  a  new  rational  func- 
tion R^ix)  which  in  in  its  lowest  terms,  and  we  shall  have 

(1)  IKx)  =  R,(x-) 

for  all  of  tliose  values  of  x  for  which  this  reduction  is  legiti- 
mate, that  is,  for  all  values  of  x  except  those  whi.cli  cause 


Art.  278]  INDETERMINATE   FORMS  461 


the  liigliest  common  factor  of  f(x)  and  gi^x)  to  assume  the 
value  zero. 

a;2  _  4 
Thus,  the  function  R(x^  — ■ 

is  equal  to  -'^iC^)  =  x  +  2 


for  all  values  of  x  except  for  x  =  2.  For  x  =  2  these  two  functions  are 
not  equal,  since  the  function  /2(x)  assumes  the  form  ()/()  for  x  =  2, 
and  therefore  is  not  defined  for  x  =  2,  while  the  value  of  Ri(x)  for  x  ~  2 
is  2  +  2  =  4. 

But  for  all  values  of  x,  with  this  one  exception,  we  have 

=  X  +  2. 

X  —  4 

If  therefore  we  allow  x  to  approach  the  limit  2,  with  the  specification 
that  X  shall  approach  2  tvithout  reaching  this  limit,  we  shall  have 

x'^ 4 

lim    —  =  liin  (x  +  2)  =  4  (see  theorem  at  end  of  Art.  2G8). 

x-^-2   X  —  2         x^l 

This  example  is  typical  of  a  large  and  important  class  of 
cases.  We  have  given  a  rational  function  R(x)  which  is 
not  in  its  lowest  terms.  Consequently  there  are  certain 
values  of  x,  which  cause  both  the  numerator  and  denomina- 
tor of  Rix)  to  vanish.  Let  x=p\ie  one  of  these  values  of 
X.  The  function  R(x)  then  assumes  the  form  0/0  for  x=p 
and    is  therefore    not  defined  for  x=p.     Consequently  the 

statement 

\m\f{x) 
\imRix)=^r>  =.^ 

x^p  lim  g{x)      0 

would  be  meaningless.  The  question  is  :  lias  R(t)  a  limit 
when  X  approaches  p,  and  if  so  what  is  the  value  of  this 
limit '? 

We  proceed  as  in  the  illustrative  example.      Reduce  R{^x) 
to  its  lowest  terms  and  let  R-^{x)  be  the  resulting  fraction. 
We  shall  have 
(1)  i2(^)  =  B,^(x) 

for  all  values  of  x  in  tlie  ueigliborhood  of  x  =  p^  excepting 
only  the  value  x  =  p  itself.     If  x  approaches  j3  in  such  a  way 


462  LIMITS  [Art.  278 

as  not  to  assume  the  value  p  during  the  approach,  we  shall 
have  (Final  Theorem  Art.  268) 

(2)  \{mR{x)  =  \imR^(x). 

But  we  have  further  either 

(3)  \\m  R^{x^  =  R^{p) 

or 

(4)  lim  R-^(x^  =  00 

x-^p 

according  as  R-^(x)^  which  is  in  its  lowest  terms,  is  con- 
tinuous in  the  neighborhood  oi  x=p  or  else  has  2;=^  as 
a  pole.  By  combining  (2)  with  (3)  or  (4)  we  obtain  the 
desired  limit. 

In  Art.  87  and  Art.  89  we  introduced  the  notion  of  the  derivative 
of  a  function.  The  variable  whicli  takes  the  j^lace  of  the  a:  of  this 
article  is  the  h  which  occurs  there.  Observe  that  we  are  there  dis- 
cussing a  problem  of  just  the  kind  treated  in  the  present  article. 
This  remai"k  will  serve  to  convince  the  student  of  the  importance  of 
such  limits. 

The  following  examples  will  illustrate  how  to  treat  other 
cases  in  which  the  direct  application  of  the  theorems  on 
limits  of  Arts.  273  and  274  give  indeterminate  or  meaning- 
less results. 

3.2  _  2 
Ex.  1.     Find  the  limit  which  - — ^ —-^ — -  approaches  when  x  grows 

beyond  bound. 

Solution.     It  is  useless  to  write 

lim(x2-2) 
lim =  ■ '-^^ °°  • 


:_^  5  x2  +  3  X  -  7       lim  (5  x^  +  3  x  -  7)      ^ 

because   5^    is  just   as   meaningless  or   indeterminate  as  0/0.     But  we 

may  write  q 

1  -- 
x^  —  2  .  ^~ 

lim  - — =  lim  5 n 

,_^  5  x2  +  3  a;  -  7      .,^^  5  4.  §  _  1 


Art.  278]  INDETERMINATE   FORMS  463 

since  the  given  fraction  is  equal  to 


1  --. 


for  all  finite  values  of  x  which  are  different  from  zero.     As  x  grows  be- 

8     '>      7 
yond  bound  -,  ^,  —  all  approach  the  limit  zero,  and  therefore 

a:2  -  2         _  1 

Ex.  2.     Evaluate  the  limit  of  (x^  +  x  -  2)  •  — -^  for  x  =  1. 

Solution.     For  x  =  1    the  given   function   assumes  the  indeterminate 
form  0  •  00.     But  we  may  write 

1  x2  +  X  -  2 


(x2  +  X  -  2) 


1  X  -  1 


which  assumes  the  indeterminate  form  0/0  for  x  =  1,  and  may  therefore 
be  treated  by  our  first  method. 


Ex.  3.     Find   lim  [L — L-      "/    f  1 
x-^2Lx2-4      2x(x-2)J 


Solution.     For  x  =  2  the   function  assumes  the  indeterminate   form 
00  —  00.     But  we  may  write 

X  -  1         2  X  -  3  -  3  X  +  6 


x2  -  4      2  x(x  -  2)      2  x(x2  -  4) 

which  assumes  the  indeterminate  form  0/0  for  x  =  2  and  may  therefore 
be  treated  by  our  first  method. 

EXERCISE  CXXVIII 
Evaluate  the  following  Ihuits. 

r2        1                                            x"  —  r;3  „               X''  —  1 

1.  lim-^^-                  4-    lim^^ ^.  7.    lim  ;;;r—r- 

,_^i  X  —  1                           x_>Kj  X  —  a  x_>.i  X-  —  1 

o     ..      .r--n-                   R     ,.      x<  -  16  «      ,.      X*  -  1 

2.  hm 5-    lim ^-  8-     lim   77-— T" 

i^.a  X-  -  a                           _>2    -r  -  J  i_>.x  •'-  +  1 

3.  hm ^-                  6.    hm 9-     bm  t-^ ■ 

x_>.2  X  -  2                                   x->.a  X  -  a  rr-^W.  2  X''  -   1 


464  LIMITS  [Art.  278 


X  +  1  T  ^  liin      1 

■  ■'■''•     I— V2 

c^  —  3  X  — 

3x3-7x2+1  T-  lim  rl 


11      r  3  x"*  -  7  x-'  +  1  3^4     iim  fl  2        H 

■    ^!^.ia:3 +5x2-7  x+l"  '   '^^  Ix      x(x  +  2)J 

12.     lim  (x2  -  1)  .  -^  .  15.     lim  ('^-±1-  nV 

x->l  -^      X  —  1  ,(->.»  V        n  / 

16.     lim   i'^^+^-n). 

,  „      lim        ax"  +  i')x"~^+  •■•  Ix  +  m 
^"^^  a'x^  +  6'x"~^  +  •••  +  /'x  +  m' 

TO      , .       rtx"  +  ix"-i  +    ■  •  ■  +  Zx  +  m 
J-o.     urn ^ — — — • 

x-^^  a'x"  +  &'x*-i  +  •••  +l'x  +  m' 

Distinguish  the  three  cases 

n  >  k,  n  —  k,  n  <C  k. 


CHAPTER   XVT 

INFINITE    SERIES 

279.  Nonterminating  geometric  progressions.      We  saw  in 

Alt.  60  that  the  sum  of  n  terms  of  a  geometric  progression 
a,  ar,  ar^,  ••.  could  be  expressed  in  the  form 

(1)  S,  =  j^-^- 
^  1  —  r      1 —r 

If  r  is  numerically  less  than  unity,  that  is,  if  |  r  |  <  1,  we  saw, 

in  Art.  275,  that 

lim  I  r"  1  =  0, 

so  that 

(2)  S  =  \\m  S„  =  -^^  if  I  r  !  <  1. 

n^-..  I  —  r 

S  is  then  said  to   be  the  sum  of  the  nonterminating  geo- 
metric progression.      (See  also  Art.  62.) 

280.  Some  other  non-terminating  series.      In  Art.  266  we 
found  the  following  formulae  (see  equations  (6)  Art. 266); 

n  1  1 

V =  1- 


^^k{k  +  -[)  71+1 

1  in  1        "I 

2!      (//  +  l)(w  +  2)J 
1  1 


f^k(k  +  l}(k  +  2)      2 
1  1 


^,k(k  +  lXk  +  2}ik  +  ^l)      8 


JV.      (n  +  l)(w  +  2)(«  +  3) 


We  conclude,  on  tlie  basis  of  the  theorems  on  limits  (Art. 
273),  that 


466 


466  INFINITE   SERIES  [Art.  281 

n  -\  "1  1 

lim  V ± =  1,    lim  V —^ =  -, 

^^  y  ^    1 _J^ 

n^^^k{k+l)(Jc  +  2)(k-]-S)      18* 
Instead  of 

lira  y  yy^^—-  =  1, 

we  usually  write 


%  K^  + 1) 


Thus  the  symbol  V  raeuns  the  same  as  lira  V. 

k=\  "-><«  A=l 

281.   Convergence   and  divergence  of   infinite   series.     We 

now  proceed  to  generalize  the  notions  encountered  in  Arts. 
279  and  280.  Let  Mj,  u^,  %,  •••  be  the  terms  of  a  non-termi- 
nating or  infinite  series,  and  let  us  denote  by  *S'„  the  sum  of 
the  first  n  terms,  so  that 

S^  =  Mj,    S^  =  U^  +  U^i    >S3  =  U^  -{-  U^  +  W3,  •»•, 

S^  =  Wj  +  y.2  +  W3  +  •••  +  "„-i  +  w„. 

Clearly  the  value  of  >S'„  will  depend,  in  the  first  place,  upon 
the  nature  of  the  series  under  consideration,  that  is,  upon  the 
law  of  formation  of  its  terms,  and,  in  the  second  place,  upon 
the  number  of  terms  included  in  S^.  We  express  this  by 
saying  that  /S'„  is  a  function  of  n. 

If  the  sum  of  the  first  n  terms  of  an  infinite  series  approaches 
a  definite  finite  limit  as  n  grows  beyond  hound,  that  is,  if 

lim  S,  =  S 

ivhere  S  is  a  definite  finite  number,  the  infinite  series 

Mj  -I-  ■?^3  +  •••  to  infinity 

is  said  to  be  convergent,  and  the  limit  S  is  said  to  be  its  sum. 

If  the  series  is  not  convergent,  /S'„  will  not  approach  a  defi- 
nite finite  limit  as  n  grows  beyond  bound,  and  we  shall  say 


Art.  281]         CONVERGENCE    AND   DIVERGENCE  467 

that  the  series  is  divergent.  There  are  two  chisses  of  diver- 
gent series ;  those  of  the  first  class  for  wliich  *S'„  becomes  in- 
finite when  n  grows  beyond  bound  ;  and  those  of  the  second 
class  for  which  *S'„  does  not  approach  a  definite  finite  number 
as  a  limit  although  8^  does  not  become  infinite.  Series  of 
the  latter  class  are  often  called  oscillating  series.  To  avoid 
confusion,  the  student  should  note  that  some  authors  use  the 
word  divergent  only  for  the  case  when  lim  >S'„  =  oo. 

The  word  sum  is  here  used  in  a  new  sense.  (Compare  also  Art.  62.) 
Our  original  definition  of  a  sum  (Art.  2)  only  applies  to  the  case  where 
the  number  of  terms  is  finite.  The  sum  of  an  infinite  series,  as  here 
defined,  is  not  a  sum  at  all  iu  the  original  sense  of  the  word;  it  is  the 
limit  which  such  a  sum  approaches  when  the  number  of  terms  grows 
beyond  bound. 

Example  1.     The  geometric  progression 

1  +  *  +  i  +  i  +  - 
is  a  convergent  infinite  series.     The  sum  of  the  first  n  terms  is 

^   -      ^  ^/^"       (See  (1),  Art.  279), 


so  that 


proving  that  the  series  is  convergent  and  that  its  sum  is  equal  to  2. 

Figure  82  illustrates  graphically  how  it  happens  that  .S",,  approaches  the 
finite  limit  2.  The  distance  OB  is  equal  to  two  units.  The  distances 
OSo,  OSz,  OSi,  and  so  on  represent  the  sum  of  two, 

three,  four,  •••  terms  of  the  series  respectively,   and     ^ '^ — >^-^' 

it  is  evident  from  the  figure  that  05„  has    OB  as  p^^  g2 

its  limit. 

We  have  seen  more  generally,  in  Art.  279,  that  any  geometric  progres- 
sion, whose  constant  ratio  is  numerically  less  than  unity,  forms  a  convergent 
series,  whose  sum  is  equal  to 

1-r 

Example  2.  Each  of  the  series  of  Art.  280  is  convergent.  Their 
sums  are  1,  i,  and  ^  respectively. 


1  - 

i 

1-i 

s,. 

_  2  _ 

On- 

i' 

lim  5„ 

=  2, 

n-^x 

468  mrmiTE   series  [Art.  282 

Example  3.  The  series  1  +  1  +  1  4-  •••  +  1  +  •••  is  divergent.  For, 
in  this  case  5„  =  n  and  therefore  S„  grows  beyond  bound  as  n  becomes 
infinite. 

Example  4.  The  series  1  —  1  +  1  —  1  +  1  —  l+--is  also  divergent. 
But  in  this  case  S„  does  not  become  infinite.  In  fact  S„  is  equal  to  zero 
when  n  is  even,  and  S,^  is  equal  to  1  when  n  is  odd.  As  n  grows  beyond 
bound,  S^  oscillates  between  the  two  values  0  and  1.  It  does  not  be- 
come infinite,  but  neither  does  it  approach  a  limit.  The  series  is  an 
oscillating  one. 

The  method  of  investigation  illustrated  so  far  is  applicable 
whenever  we  actually  know  how  to  find  an  exact  and  simple 
expression  for  S„^  the  sum  of  the  first  n  terms  of  the  given 
series.  This  is  one  of  the  reasons  why  it  is  a  matter  of  great 
importance  to  be  able  to  find  the  sum  of  a  finite  series.  (See 
Arts.  262  to  266.)  Moreover  in  all  such  cases  we  can  do 
more  than  merely  decide  the  question  as  to  whether  a  series 
is  convergent  or  divergent.  If  it  is  convergent,  we  can  ac- 
tually find  its  sum. 

EXERCISE  CXXIX 

Discuss  the  convergence  or  divergence  of  the  following  series,  and  find 
the  sum  if  the  series  is  convergent. 

2.  1+2  +  4  +  8+....  ■    1-2-3      2.3-4      3.4-5 

3.  •_)  _  2  +  2  -  2  +  2  ...  .  7.    1  .  2  +  2  •  3  +  3  •  4  +  ...  . 

4-    7  +  1  +  J  +  I  +  ^'e-  +  ... .  8.    1  +  3  +  5  +  7  +  ...  . 

5.    -L  + J_+ J_+  ....  9.2  +  4  +  6  +  8  +  .... 

1.22-33.4 

282.  Fundamental  criteria  for  convergence.  Whenever  we 
have  no  explicit  formula  for  *S'„  we  cannot  decide  whether  a 
series  is  convergent  or  divergent  by  the  method  of  Art.  281. 
We  must  therefore  seek  for  more  general  methods. 

Let  the  series 

(1)  u^  +  ^2  +  ^3-1-  ••• 

be  convergent.     Let  S  be  the  sum  and  let  S^  be  the  sum  of 
its  first  n  terms.     Then  >S'  —  *S'„  must  approach  the  limit  zero 


Art.  282]  FUNDAMENTAL   CRITERIA  469 

when  )i  grows  beyond  bound,  and  the  same   thing  must  be 
true  of  S—Sn-y     Consequently  the  difference 

must  also  approach  zero  as  a  liuiit.      But 

lS'„  =   Wj  +   Wg  -+-    •  •  ■    +  Wn-1+  "n^ 

SO  that  S^  —  Sn-i  is  equal  to  m„.     Consequently  we  obtain 
the  following  result. 

Theore^ni  I.  If  a  series  is  convergent  its  nth  term  must 
approach  the  limit  zero ^  when  n  grows  beyond  hound;  that  is ^ 
the  condition 

(2)  lim  M„  =  0 

must  he  fulfilled  if  the  series  (1)  is  to  he  convergent. 

The  condition  (2)  is  necessary  for  convergence^  hut  hy  no 
means  sufficient.  In  other  words,  it  may  happen  that  con- 
dition (2)  is  satisfied  and  that  the  series  is  nevertheless 
divergent. 

Such  is  the  case,  for  instance,  for  the  following  series,  the  so-called 
harmonic  series, 

(3)  i  +  J  +  i+l+-. 
In  this  case  we  have 

w,^  =  —  and   lim  Un  =  liin  -  =  0, 
n  «->-»  ?»->«  n 

but  the  series  is  nevertheless  divergent,  as  we  shall  now  show. 
We  may  write  (3)  as  follows : 

(4)  1  +  ^  +  (1-  +  \)  +  {\  +  i  +  ^  +  1)  +  (i  +  A  +  •••  +  h)  +  ■■; 
where  the  terms  are  collected  in  groups  of  one,  two,  four,  eight,  and  so 
on  in  accordance  with  a  law  which  is  easily  recognizable.     Not  counting 
the  first  term  at  all,  the  first  group  consists  of  a  single  term  1/2.     The 
sum  of  the  terms  of  the  second  group  is 

I  +  \,  which  is  greater  than  ^  +  ^  or  ^. 
The  sum  of  the  terms  of  the  third  group 

^  +  I  +  J  +  ^  is  greater  than  '  +  J  +  1  +  i,  «•  e.  greater  than  \. 


470  INFINITE   SERIES  [Art,  282 

The  mth  group  consists  of  2'""^  terms,  namely 


Each  of  these  terms  is  greater  than  the  last,  whose  value  is  equal  to 
1  ^       1        ^  1 

Om-l  _|_   >2m"l  ~~  2  •  2"'~1  ~  2™  * 

The  sum  of  the  terms  of  the  mth  group  is,  of  course,  greater  than  the  num- 
ber of  these  terms  multiplied  by  the  value  of  the  smallest  one  among 
them,  that  is,  greater  than 

2m         2 

Consequently,  if  we  denote  by  Sm  the  sum  of  those  terms  of  (4)  which 
are  included  in  the  first  m  groups,  we  have 


(5)  .S„  >  1  + 


o' 


since  the  sum  of  the  terms  in  each  group  is  greater  than  1/2.  But 
according  to  (5),  Sm  becomes  infinite  when  m  grows  beyond  bound,  that 
is,  when  n  becomes  infinite.     Therefore  the  series  (3)  is  divergent. 

We  have  seen  that  the  condition  (2)  is  necessary,  but  not 
sufficient  for  the  convergence  of  an  infinite  series.  The  fol- 
lowing theorem  gives  a  condition  which  is  both  necessary 
and  sufficient ;  but  we  shall  merely  state  this  theorem  with- 
out proof  since  the  proof  is  a  little  difficult  for  a  beginner. 

Theoreisi  II.     In  order  that  the  series 

Wj  -f  ^2  +  ••• 
may  he  convergent,  it  is  necessary  and  sufficietit  that,  not  only 
the  nth  term,  but  the  sum  of  any  number  of  terms  following  the 
7ith  term  shall  approach  the  limit  zero  when  n  grows  beyond 
hound. 

EXERCISE    CXXX 
Prove  that  the  following  series  are  divergent. 
1.    1  +  2  +  4  +  8+  ••.. 


2;(-^)- 


2.   2  +  2  +  2  +  2+  -.  ^  _j_ 


3.   (l  +  |)  +  (l  +  :)  +  (l  +  i)  + 


lOUO 


*  For  the  significance  of  this  notation  see  Art.  264  and  end  of  Art.  280. 


Arts.  283,  284]  COMPARISON   TESTS  471 

283.  Series  all  of  whose  terms  are  positive.  We  shall 
assume  the  following  theorem  without  proof.  The  student 
will  easily  convince  himself  of  its  great  plausibility. 

Theore:si  I.  If  all  of  the  terms  of  a  series  are  positive,  it 
cannot  he  an  oscillating  series.  It  is  either  convergent,  or  else 
the  sum  of  the  first  n  terms  will  become  infinite  as  n  groivs  be- 
yond bound. 

We  may  also  formulate  this  statement  as  follows. 

Theore^f  II.  An  infinite  series  of  positive  terms  is  con- 
vergent if  Sn  remains  finite  for  all  values  of  n,  that  is,  if  there 
exists  a  finite  positive  number  M,  such  that  >S'„  <  M  for  all 
values  of  n,  no  matter  how  great. 

For  according  to  Theorem  I,  such  a  series  is  either  con- 
vergent, or  else 

lim  >S'„=  30. 

But  the  latter  possibility  is  excluded  if  >S'„  <  J/ for  all  values 
of  n. 

EXERCISE    CXXXI 

1.  Formulate  the  theorems  which  correspond  to  Tlieorem  T  and  11  in 
tlie  case  of  series  all  of  whose  terms  are  negative. 

284.  Comparison  tests.  The  method  outlined  in  the  fol- 
lowing theorem  often  enables  us  to  prove  the  convergence  of 
a  given  series,  by  comparing  it  with  another  series  whose 
convergence  has  been  established  previously. 

Theorem  I.     Let  it  be  known  that  the  series  of  positive  terms 

(1)  ^1+  ^2  +  ^'3  -•-•••+   ^'n+     ••• 

is  convergent,  and  let 

(2)  ?/l  +  W2  +  W3+    •••    +Mn+    ••• 

be  a  second  series  of  positive  terms,  ivhose  convergence  is  to  be 
tested.     If 

(3)  w„  ^  v„, 


472  INFINITE   SERIES  [Art.  285 

for  all  of  those  values  of  n  which  follow  a  certain  first  value  of 
n  for  which  the  inequality/  (3)  is  fulfilled,  the  series  (2)  is  con- 
vergent. 

Proof.  Since  the  series  ?'i  +  re  +  •••  is  convergent,  the  sura  of  any 
number  of  its  terms  following  its  nth.  term  will  approach  zero  as  n  grows 
beyond  bound.  (See  Theorem  II,  Art.  282.)  Therefore  we  can  make 
the  sum  of  the  p  terms  which  follow  r„,  that  is, 

»"n+l  +   ''n+2  +    ••■    +  Vn-^p, 

arbitrarily  small  by  choosing  n  large  enough.  (See  Art.  268,  definition 
of  a  limit.)  We  may  also  assume  that  »i  has  been  chosen  so  large  as  to 
insure  at  the  same  time  the  validity  of  all  of  the  inequalities 

We  shall  then  have 

Wn+l  +  Un+2  +    ••■    +  M„+p  <  l'„+l  +    V„+2  +    •"    +  Vn+p. 

But  this  means  that  m  +  U2  +  •■■  +  u„+p  may  be  made  arbitrarily  small  by 
choosing  n  large  enough.  In  other  words,  the  sum  u„+i  +  u„+2  +  •••  +  «n+p 
will  approach  the  limit  zero  as  ?i  grows  beyond  bound,  no  matter  how 
large  or  small  j3  may  be.  But  this  means  that  the  series  wi  +  W2  +  •••  is 
convergent.     (See  Theorem  II,  Art.  282.) 

Theorem  II.  Let  the  series  of  positive  terms  v^  +  v.^+  ■•• 
he  divergent,  and  let 

II    ^  v 

"n  ^   '^n 

for  all  of  those  values  of  n  which  follow  a  certain  first  value  of  n 
for  which  this  inequality  is  fulfilled.  Then  the  series  u-^  +  u^-\-  •■• 
is  also  divergent. 

For,  if  this  were  not  so,  that  is,  if  u^  +  u^  +  •••  were  con- 
vergent, according  to  Theorem  J,  v-^  +  v^+  •••  woukl  also  be 
convergent  contrary  to  our  assumption. 

285.  Some  convenient  comparison  series.  In  order  to  be 
able  to  apply  Theorems  I  and  II  of  Art.  284,  it  is  necessar}' 
to  have  some  series  at  our  disposal  whose  convergence  or 
divergence  has  already  been  established.  We  have  found 
some  such  series  already.  (See  Arts.  279  and  280.)  More- 
over,  as  soon  as  we  have  proved    some  new  series  to  be 


Art.  285]        SOME  CONVENIENT  COMPARISON  SERIES        473 

either  convergent  or  divergent  by  this  method  we  may  make 
use  of  it  for  the  purpose  of  examining  still  other  series. 

The  following  theorem  is  particularly  useful  in  connection 
with  the  comparison  tests. 

The  series 

(1)  T-  +  ^  +  ^  +  -r+-  +-+••• 

iP      2p      8p      4p  7t" 

iH  convergent  wlien  p  >  1.  It  is  diveryent  when  p  =  1,  or  ichcn 
p  <  1. 

Proof.  We  have  already  shown  that  this  series  is  divergent  for  />  =  1 
since  in  tliat  case  it  reduces  to  the  harmonic  series 

(2)  1+1  +  14-1+  ...   +1+.... 

L:      o      4  n 

(Compare  Art.  282.) 

if  yj  <  1,  we  iiave  n'><.n  for  all  values  of  n  except  for  n  —  1.  (See 
Theorem  V,  Art.  1(>3.)     Therefore 

->1,  for  n  =2,3,  1.  •••, 

nP       n 

so  that  Theorem  II  of  Art.  281  assures  us  that  the  series  (1)  is  divergent 
for  p  <  1 . 

It  remains  only  to  show  that  (1)  is  convergent  when  p  >  1.  To  do 
this  we  arrange  the  terms  of  (1)  in  groups  of  two,  four,  eight,  and  so  on, 
as  we  did  in  Art.  282  for  the  harmonic  series. 

We  have 

2p     ;ip     2p     2p-i' 

4p       op      Op      7p       4p      4p~^ 

1  +  i-  -L  <ii=  J_ 


From  these  iuequalities  we  conclude.  l)y  addition, 

('i>,\  -!-  +  _  +  _+  ...  <'_J__  +     ^     +    ■*■    +  ... 

^*  ^  ^p^p^p  2P-1     <^p-\      %p-^ 

But  the  right  member  of  (3)  is  a  geometric  progression  whose  first  term 
is  a  =  1/2''-^  and  whose  common  ratio  is  r  =  1/2''-^  This  progression 
is  a  convergent  series  if  the  common  ratio  is  less  than  unity,  that  is,  if  p 


474  INFINITE   SERIES  [Art.  286 

is  greater  than  one.     Therefore,  the  series  in  the  left  member  of  (3)  is 
convergent  if  jo  >  1,  as  was  to  be  proved. 

We  may  even  draw  a  further  conclusion.     The  sum  of  the  non-termi- 
nating geometric  progression  in  the  right  member  of  (3)  is 

1 

n  OP-1  1 


1-r      i__L      2*^1 -1 

OP-l 

Therefore  we  see  that 

(4) 
if;j>l. 

EXERCISE    CXXXII 
Examine  the  following  series  for  convergence  or  divergence. 

1.    1 +i  +  l  +  i+ ... +i-+  ..., 

2^      33     44^       ^„«^ 

2      3  •  2      4  .  22      5  .  23  (n  +  l)2"-i 

3.    1 +_L.  +  ^_  +  ^_^  ...  +_J_+ .... 
2  .  22      3  .  32     4-42  n.  71- 

4. .  1  +  _L  + -L  +  J- +  ...  +  J- +  ... . 

v'2      V3      V4  Vn 

i^  8.    2;ifor.<l. 

^1.  9.     y%forO<3:<l. 

n=l  "    ^ 

7      y  £!  for  0  <  x  <  1.  10.    y    T^VTT  for  0  <  :c  <  1. 

'•     A  7i-  ^H(n+1)  ^ 


n=l        71 

6. 

7i  =  l 


n=l 


286.  Ratio  test.  Of  all  of  the  tests  which  we  shall  give 
in  this  book,  the  following  is  the  most  important. 

Theorem  I.  Let  u^  +  n^  +  u^  +  •••  he  an  infinite  series  all 
of  whose  terms  are  positive^  and  let  us  consider  the  ratio  of  the 
(n  +  lyth  term  to  the  nth  term  of  the  series,  that  is,  the  ratio 
Wn+i/**n-  ^  ^^^'^  ratio  approaches  a  limit  ivhen  n  groivs  beyond 
bounds  and  if  lim  m„+i/w„  <  1,  the  series  is  convergent. 


Art.  286]  RATIO  TEST  475 

If  lim  w„+i/?/„  >  1,  tlie  series  is  divergent.      The  question 

of  convergence  or  divergence  remains  undecided  by  this  test,  if 
either  lim  m„+i/?/„  =  1,  or  if  Un+i/Un  does  not  approach  any 

definite  limit  as  n  groivs  beyond  hound. 

Proof.     Let  us  consider  first  the  case  when 
lim  ^^2+1  <  1. 

»!—><»      Uy^ 

Let  us  denote  by  r  the  limit  of  w„-f.i/M„,  so  that 
(1)  lim^*2+l  =  r<  1. 

According  to  the  definition  of  a  limit,  tlie  meaning  of  (1) 
may  also  be  stated  as  follows.  Let  us  consider  the  sequence 
of  ratios 

-^■,    -^,    -^,    '•'■, 

W'-f  Wrt  It'O 

and  let  us  choose  a  positive  number  S,  which  may  be  taken 
as  small  as  we  please.  Then  there  will  present  itself  sooner 
or  later  a  first  one  of  these  ratios,  say 


which  differs  from  r  by  less  then  8,  and  such  that  w„+i/m„  for 
all  values  of  n  which  are  greater  than  m  will  also  differ  from 
r  by  less  than  8. 

Thus,  after  B  has  been  chosen,  m  can  be  determined  in  such 
a  way  that  all  of  the  ratios 

*,        ■ «,     ...,      ',      ... 

will  be  included  between  r  —  8  and  r  +  8.  Since  r  was,  by 
hypothesis,  less  than  unity,  and  since  8  was  a  positive  number 
which  could  be  chosen  arbitrarily  small,  we  may  in  particular 
choose  8  in  such  a  way  that  r  -\-  8  will  also  be  less  than  unity. 


476  INFINITE   SERIES  [Art.  286 

In  Fig.  83  the  line-segment  OU  is  one  unit  long,  and  07?  =  r  rep- 
resents the  limit,  less  than  unity,  which  u,^^Ju^  approaches  as  n  grows 
beyond  bound.     If  8  is  chosen  as  a  positive  number 

f\  J        Tf      "K     XJ 

^ ^i»  r  r+l  \       such  that  ?•  -f  S  is  still  less  than  unity,  the  line-seg- 

„      „„  ments  OL  and  OK  represent  the  numbers  ?•  —  8  and 

Fi<3.  83  „  . 

r  4-  o  respectively.  The  statement  that  «„  n/",,  ap- 
proaches r  as  a  limit  is  equivalent  to  saying  that  the  various  line-segments 
which  represent  the  quotients  M„r]/w„  for  growing  values  of  n  will  ap- 
proach OR  as  a  limit,  so  that  if  m  is  taken  large  enough,  all  of  the 
quotients  w„+i/w„,  for  which  n  exceeds  m,  will  lie  between  OL  and  OK. 

Now  let  us  put  r  +  S  =  A".     Then  A;  <  1,  and  we  have 
(2)  ?^^tt2<^A-, 


m+l 


^^A;, 


'm+2 


and  so  on.     But  from  (2)  we  find 


■^m+i  ^  "^w„ 


\6)  "MTO+ajS-^^m+i^^  ^TO? 

and  so  on.     If  we  compare  the  sum  of  the  left  members  of 

(3)  to  the  sum  of  the  right  members,  we  conclude 

(4)  w,„+j  -f  w,„+2  +  ^™+3  +  •  •  •  :4  w^(^  +  A'-  +  F  -f  •  •  •). 

Since  k  is  less  than  unity,  the  geometric  progression  in  the 
right  member  of  (4)  is  convergent.  Consequently  the  series 
Wm+i  +  w„+2  +  '■■  ^^  convergent.  If  we  add  to  this  series  the 
first  7W  terms  u^  +  u^-\-  ■■•  -}-  w„,  we  alter  the  value  of  the  sum, 
but  the  series  will  remain  convergent. 

Thus  we  have  proved  that  our  series  u^-\-u<^-\-  •••is  con- 
vergent if 

lim  ^^^  <  1. 

If  instead 

lini  !^2+l  =  r  >  1, 

n— >.oc    W„ 


Art.  286]  RATIO   TEST  477 

a  slight  modification  of  our  argument  shows  that  vi  may  be 
chosen  so  large  that  u,n+\/'^m  ^^^  3-^1  of  the  ratios  m„+i/w„  for 
which  n  >  m  will  be  greater  than  unity. 

In  Fig.  83  the  point  R  would  be  to  the  right  of  U  and  both  points  L 
and  K  will  be  to  the  right  of  [/  if  S  is  chosen  sufficiently  small. 
Thus  we  shall  have  in  this  case 

and  so  on,  so  that 

liroving  that  w^  +  Wj  +  •••  must  be  divergent  if 

lim  ^fii±i>l. 

n— >«      W„ 

This  last  argument  enables  us  to  complete  our  theorem  by  the  follow- 
ing statement : 


If 


lim  ^^5+1  =  ^  =  1, 


hut  if  all  of  the  ratios  of  the  sequence 


*'TO+l         "'m+2         "'m+3 
Urn  Um^  I        Ufn+2 


which  folloiv  a  certain  first  one  u^j^jum^  are  greater  than  or 
equal  to  unity,  then  we  may  still  assert  that  the  series  is 
divergent. 

But  the  ratio  test  gives  us  no  information  whatever  in  case 
lim  ?^^  =  r  =  1, 

n->oo    Un 

while  all  of  the  ratios  of  the  sequence 

are  less  than  unity.  Such  a  series  may  be  convergent  or 
diverofent. 


478  INFINITE   SERIES  [Art.  286 

Thus  the  harmonic  series 

'J      o  n 

is  known  to  be  divergent.     In  this  case  we  have 


M„        n  +  \      n      n  +  \      1j-1 

n 

This  ratio  is  less  than  unity  for  all  values  of  n,  but  it  approaches  unity  as 
a  limit.     The  reason  that  the  proof  of  convergence  for  the  case 

lim  ^/i+i  —  ^^1 

is  not  applicable  to  this  series  is  easily  seen.     Although  we  have 

!f»±i<l 

for  all  values  of  n,  we  cannot  assert  that  there  exists  a  fixed  number 
A;<  1,  the  same  k  for  all  values  of  n,  such  that 

In  fact,  since  Un+Ju,^  appi'oaches  1  as  a  limit,  we  know  that  no  such 
number  k  exists  which  is  less  than  1.  Consequently  the  argument  based 
upon  the  inequalities  (2),  (3),  and  (4)  fails  to  prove  convergence  of  the 
series  since  the  lowest  value  of  k  which  we  can  use  in  this  case  is  ^  =  1, 
and  not  a  value  of  k  which  is  less  than  1.  But  for  A-  =  1  the  series  in 
the  right  member  of  (4)  is  divergent,  and  consequently  the  inequality  (4) 
fails  to  prove  that  the  left  member  is  a  convergent  series. 

It  may  happen,  however,  that  a  series   is  convergent   although  the 
limit  of  u„^.j/w„  is  equal  to  unity.     This  is  the  case  in  the  series 

We  proved  in  Art.  285  that  this  series  is  convergent.     But  we  have 

^5+1=         1  .    1  ^        n2        ^  n^  1 

M„        {n  +  iy-    '  n^~  {n  +  \y^       n2  +  2  ?i  +  1       j  ^  2  ^  J_ 

n       n^ 
and  this  approaches  unity  as  a  limit  when  n  grows  beyond  hound. 

Thus  we  have  the  following  result. 

Theorem  II.     Ifu^^Ju^  does  not  tend  toivard  a  definite  finite 
limit  or  if  lim  Ur^Ju^  =  1,  then  the  ratio  test  fails  to  decide  the 


Art.  287]        RATIO   OF   TERMS   OF   TWO   SERIES  479 

question  zvhetker  the  series  of  positive  terms  u^  +  U2+  u^+  ■■■ 
is  convergent  or  divergent. 

Of  course  such  a  series  will  actually  be  either  convergent  or  divergent. 
We  are  merely  asserting  that  the  ratio  test,  in  such  cases,  is  powerless  to 
decide  the  question  and  that  a  decision  must  be  souglit  by  some  other 
method. 

The  limit    lim  w„+i/?^„  =  /'  is  often  called  the  test  ratio,  al- 

though  it  is  really  no  ratio  at  all,  but  the  limit  which  a  cer- 
tain ratio  approaches  when  n  grows  beyond  bound. 

EXERCISE    CXXXIII 
Apply  the  ratio  test  to  the  following  series: 

1.  1+1  +  1+1+....  6.    1  +  -1-+1+.... 

1 !      2 !      3 !  1 !      3 !      .'j ! 

2.  1+1+21+23  _  7.    1+1  +  ^+  .... 

1 !      2  !      3  ! 

3.  1+1  +  ^+^+....  8.    1  +  1  +  1+  .... 

1 !      2  !      3  ! 

4.  1  +  £  +  ?i  +  ^  +  ....  9.    1+2+4  +  8  +  .... 


1 !      2 !      3  ! 

:i\,      ^['^  ■"■  100  '  100^   '   1008 


02       32      42  ,„      2!    ,     3!     ,     4!     , 

5     l4.n_L_-L_-L...  10. h 


287.  Ratio  of  corresponding  terms  of  two  series.  The  fol- 
lowing theorem  is  convenient  in  many  cases. 

Let  the  ratio  of  the  nth  terms  of  two  series  of  positive  terms 
he  finite  and  different  from  zero  for  all  values  of  n,  and  let  this 
ratio  approach  a  limit,  ivhen  n  groivs  beyond  bound,  which  is 
also  finite  arid  different  from  zero.  Tfien  either  both  series  are 
convergent,  or  else  both  are  divergent. 

Proof.     Let 

(1)  Ui  +  ^2  +  W3  +    •  •  •    +  Un  +  •  •• 

and 

(2)  v^  +  v^  +  v^+  ...  +t^„+.-. 

be  the  two  series  under  consideration,  every  term  of  each 
series  being  positive.     Let  us  assume  that  M„/y„  is  finite  and 


480  INFINITE   SERIES  [Art.  287 

different  from  zero  for  all  values  of  n  and  that  this  ratio 
approaches  a  limit 

lim  ^  =  k, 

which  is  finite  and  different  from  zero.      We  wish  to  prove 
tliat,  under    these  conditions,  (1)  and  (2)  are  either  both 
convergent  or  else  both  divergent. 
Put 

(3)  Sn  =  U^  +  ^2  +    •  •  •   +  M.„, 

(4)  ASV   =  Vl  +  f2+    •••    +    ^'n, 

and  let 

According  to  our  hypothesis,  r j,  r^,  •••  r„  are  finite  positive 
numbers,  all  different  from  zero.  Let  us  denote  by  r„'  the 
smallest  and  by  R^  the  greatest  of  these  n  numbers.  Then 
we  shall  have 

or 

(6)  rjv^  ^  ^1  ^  ^n'^r  •  •  •'  ^nVr,  ^  W„  ^   RJVr,. 

From  these  inequalities  we  find  by  addition 

or,  making  use  of  (3)  and  (4), 

(7)  r^S^^S^SRuSr!. 

The  values  of  r„'  and  RJ  will  ordinarily  be  different  for 
different  values  of  n.  But,  according  to  our  assumptions, 
all  of  these  numbers  will  be  finite  and  different  from  zero. 
Consequently  there  will  exist  two  finite  numbers  different 
from  zero,  r  and  72,  such  that  the  inequalities 

(8)  r^r^  and  R^Rr! 
will  be  satisfied  for  all  values  of  n. 


Art.  287]        RATIO  OF  TERMS  OF   TWO   SERIES  481 

The  ouly  thing  wliich  could  interfere  with  the  existence  of  two  such 
numbers  would  be  the  possibility  that  w„/r„  might  approach  the  limit 
zero  or  become  infinite.  But  both  of  these  possibilities  are  excluded 
by  the  hypothesis  that  this  limit  should  be  finite  and  not  zero. 

From  (7)  and  (8)  we  can  conclude 


whence,  since  SJ  is  positive  and  surely  not  equal  to  zero, 

(9)  r<§^,<R. 

Thus,  the  ratio  of  S^  to  SJ  will  lie  between  the  two  positive 
numbers  r  and  M  for  all  values  of  n. 

Since  all  of  the  terms  of  (1)  are  positive,  (1)  cannot  be 
an  oscillating  series.  Consequently  *S'„  will  either  approach 
a  finite  limit  as  n  grows  beyond  bound,  or  else  S„  will  be- 
come infinite.  (See  Theorem  I,  Art.  283.)  The  same  thing 
is  true  of  *S'„'.  Since  *S'„'  cannot  have  zero  as  a  limit,  every 
one  of  its  terms  being  positive,  we  have 

a        lim  *S'„ 

„_^^  SJ       lim  *S'„' ' 

n-<— 00 

and  therefore  we  conclude  from  (9), 

lim  *S'„ 

aO)  r^r^^^^—-<B. 

lim  .S„'  — 

n   >  j: 

If  *S'„  has  a  finite  limit,  the  same  thing  must  be  true  of  SJ. 
For,  as  we  have  just  noted,  S^  either  has  a  finite  limit  or 
becomes  infinite,  and  the  latter  possibility  would  contradict 
the  inequality  (10).  A  similar  contradiction  would  arise  if 
lim  *S'„  =  X  unless  we  have  also  lim  S„'  =  ^. 

Thus  our  theorem  is  proved. 

EXERCISE    CXXXIV 
Examine  the  convergence  or  divergence  of  tlie  following  series: 
1-    ]  +|  +  i  +  i  +  -- 
2.    i  +  i  +  i  +  l  +  .... 


482  INFINITE    SERIES  [Art.  288 

3. 1 [-  —  +  —  +  •••.     Distinguish  the  cases  ^' >  1  and  A:  <  1. 


Ifc       3k       5fc       7A; 


•  ^  7  +  5  n  ^  /i^  +  8  n2  +  4  n  +  1 

i.  y    •'^^-^  "  7.  y 

^  5  n2    I    7  „  _  1  A' 


5?i2  +  7„_l  frinVn  +  S 

288.  Series  with  positive  and  negative  terms.  Clearly  the 
criteria  which  we  have  developed  for  series  all  of  whose 
terms  are  positive  are  applicable,  with  very  minor  changes, 
to  series  all  of  whose  terms  are  negative,  or  to  series  all  of 
whose  terms,  excepting  only  a  finite  number,  have  the  same 
sign.  We  may  make  use  of  these  criteria  also  for  many 
series  which  contain  an  infinite  number  of  terms  of  either 
sign,  as  a  consequence  of  the  following  theorem. 

Theorem  I.  An  infinite  series  u^  +  u<^+  %+  •••  whose  terms 
are  all  real,  but  not  necessarily/  all  positive,  will  be  convergent, 
if  the  series  of  positive  terms, 

composed  of  the  absolute  values  of  the  terms  of  the  original 
series,  is  convergent. 

Proof.    Let 

/^■v  <S„  =  Ml   +  «2  +    •■•    +  M„. 

2„  =|in|  +  |r<2|  +  •••  +1  w„I- 

The  sum  of  any  number  of  terms,  say  p  terms,  of  wj  +  W2  +  •••  which 
follow  the  ?ith  term  m„  is 

(2)  W„  +  l   +   "n+2  +    •••    +  Un+p. 

The  numerical  value  of  this  sum  will  be  smaller  than  or  at  most  equal  to 

(3)  |Wn+l!+  |Mn  +  2(+    •••    +  I  "n+p|, 

since  the  corresponding  terms  of  the  two  sums,  (2)  and  (3),  are  numeri- 
cally equal  to  each  other,  and  since  all  terms  of  (3)  are  positive  while 
(2)  may  contain  positive  and  negative  terms.  But,  whatever  value  p 
may  have,  the  sum  (3)  will  approach  the  limit  zero  as  n  grows  beyond 
bound,  since  the  series 

|wi  l  +  l  W2I+  ••• 


Art.  288]       ABSOLUTELY   CONVERGENT  SERIES 


483 


is  convergent  by  hypothesis.  (Theorem  II,  Art.  282.)  Consequently 
the  sum  (2),  whose  numerical  value  is  at  most  ecjual  to  that  of  the  sum 
(8),  will  also  approach  the  limit  zero  as  n^  grows  beyond  bound.  But 
this  means  (Theorem  II,  Art.  282)  that  the  series 

Ul  +  U2+U3  +   ••■ 

is  convergent,  as  was  to  be  proved. 

A  series  of  this  sort,  wliich  is  not  merely  convergent,  but 
which  has  the  further  property  tliat  the  series  composed  of 
the  absolute  values  of  its  terms  is  also  convergent,  is  said  to 
be  absolutely  convergent. 

The  ratio  test  (see  Art.  286)  may  now  be  extended  so  as 
to  become  applicable  to  series  whose  terms  are  not  all  positive. 

Theorem  II.  The  series  Ui  +  U2+  •••,  whose  terms  need 
not  all  have  the  same  sign,  is  convergent  if 


It  is  divergent  if 


lira 

W„+i 

7i->00 

Un 

lim 

Wn+l 

n-^o: 

Wn 

<1. 


>  1. 


The  test  fails  to  give  any  information  if 


lim 


=    1. 


Proof.  The  first  assertion  is  an  immediate  consequence  of  Theorem  I. 
The  second  follows  from  the  fact  that  the  limit  of  «„  cannot  be  equal  to 
zero  if 


lim 

9i— ^<0 


Wn+1 


>1, 


and  we  have  seen  (see  Art.  282)  that  in  any  convergent  series  the  limit 
of  u,i  must  be  zero.  The  third  assertion  is  merely  a  reiteration  of  what 
we  found  before.     (Theorem  II,  Art.  286.) 


EXERCISE   CXXXV 

The  following  series  are  to  be  tested  for  convergence  or  divergence 

-+  .... 


1!      2!      ;]!      4!      5! 


1! 


■L  +  1_JL  + 

3  !      5 !      7  ! 


484  INFINITE    SERIES  [Arts.  289, 290 


3. 

1        11        1 
2!      4!      6!      8! 

4. 

1  -.  1  +  i  -  i  +  • 

5. 

1-1+^-1+ 
22  ^32     42  ^ 

289.  Conditionally  convergent  series.  A  series  with  posi- 
tive and  negiitivo  terms  may  be  convergent  although  it  is 
not  absolutely  convergent.  (See  the  definition  of  absolute 
convergence  in  Art.  288.)  Such  series  are  said  to  be  con- 
ditionally convergent.  A  simple  illustration  of  a  condition- 
ally convergent  series  will  be  given  in  the  next  article. 

290.  Alternating  series.  An  alternating  series  is  one 
whose  terras  are  alternately  positive  and  negative.  If  Wj,  Wgi 
Wg,  W4,  •••  denote  positive  numbers, 

(1)  Wj  —  W2  +  Wg  —  M4  -H  W5  —  Wg  -I 1 •  •  • 

is  an  alternating  series.  Any  alternating  series  may  be 
expressed  either  in  the  form  (1)  or  else  in  the  form 

(2)  — (Wj  —  M2  +  Wg  — 7/4-I —  ••■). 

Since  (1)  and  (2)  are  convergent  or  divergent  at  the  same 
time,  it  sujBfices  to  consider  a  series  of  form  (1).  We  have 
the  following  theorem  due  to  Leibniz  : 

An  alternating  series  is  convergent  if  each  term  is  numericalli/ 
less  than  the  preceding  one,  and  if  the  nth  term  of  the  series 
approaches  the  limit  zero  when  n  becomes  infinite. 

PuooK.  Let  j/j.  u^,  «..  •••  be  positive  numbers,  such  that 
(8)  Uj>W2>  "3>  "4>---  >u„>  w„+i>  •••, 

and  let 
(4)  lim  w„  =  0. 

n->co 

If  n  is  an  even  number,  we  may  write 

(o)  S,^  -{ui  -  M,)  +  (M3  -  M4)  +  •••  +  (w„_i  -  u„) 

where  each  parenthesis  is  positive  on  account  of  (.3),  so  that  S„  is  surely 
]iositive.     We  may  also  write 

(6)  S^  =  "1  -  (",,  -  W3)  -  ("4  -  W5) ("«-2  -  "«-i)  -  ""• 


Art.  291]  SERIES  OF  FUNCTIONS  485 

Again  each  of  the  differences  inclosed  in  a  parenthesis  is  positive,  so 
that  (6)  tells  us  that 

(7)  S„<u,. 
Tlius  the  series  whose  terms  are 

Ui  —  w,,   M3  —  W4,    J'a  —  Uf.,  ■•■ 

has  all  of  its  terms  positive,  and  there  exists  a  positive  finite  number, 
namely  Ui,  such  that  the  sum  of  any  number  of  terms  of  this  series  is  less 
than  Ml-  Consequently  (see  Theorem  II,  Art.  28:3),  tliis  series  of  posi- 
tive terms  is  convergent. 

In  other  words,  the  sum  of  an  even  number  of  terms  of  our  alternating 
series  approaches  a  definite  finite  limit  S,  if  the  number  of  terms,  always 
remaining  even,  grows  beyond  bound.     Thus,  if  n  is  even,  we  have 

(8)  lim   S„  =  S. 

n— ><» 

But  if  n  is  even,  n  —  1  is  odd,  and  we  have 

Consequently 

lim   5„_i  =  lim   S,^  —  lim   ?/„  =  S 

n— >.<»  n — >-»  n    >oo 

on  account  of  (4)  and  (8).  Thus  5,,  approaches  the  same  definite  finite 
limit  S  whether  n  be  even  or  odd,  and  therefore  the  alternating  series  is 
convergent. 

EXERCISE  CXXXVI 

Investigate  the  following  series  for  convergence  or  divergence : 

-^     C—  l)"n 

n=l 


\ L+    1  _     ^     ,  .   v^(-i)"iO". 


^(-1)"10' 
*•    jLf    10"  +  n 


l.I      1.02      1.00:3      1.0001  ^    10"  + 

,1=1 

5.   I-L4--I---L  +  -L-.... 
\/2      v';3      Vl      V5 

291.   Series  whose  terms  are  functions  of  x.      Let  ns  con- 
sider a  series 

v^iz)  +  u.^(y)  +  u^{x)  4-   ••• 

wliose  terms  are  fuiictioiis  of  x.  If  we  i)ut  for  x  some  par- 
ticular value,  such  as  x  =  a  or  x  =  b^  we  may  examine  tlie 
convergence  of  the  series  in  each  of  these  cases.  It  may 
happen  that  such  a  series  is  convergent  for  some  values  of  x 


486 


INFINITE  SERIES 


[Art.  292 


and  divergent  for  others.  If  it  is  convergent  for  two  differ- 
ent values  of  x^  x  =  a  and  x  =  b,  we  may  expect  it  to  have 
different  sums  in  the  two  cases. 

All  of  those  values  of  x  for  which  a  series  of  the  form  (1) 
is  convergent,  are  said  to  constitute  its  domain  of  conver- 
gence. For  all  values  of  x  in  its  domain  of  convergence,  the 
series  defines  a  function  of  x. 

292.  Power  series.  The  simplest  and  most  important 
case  of  this  kind  is  that  of  a  power  series 

Uq -{- a^x -\- a^x^  +  •  •  •  -H  «„a;" -|-  •••, 

every  term  of  which  is  a  product  of  a  constant  a„  (whose 
value  depends  upon  n  but  not  upon  x}  multiplied  by  the 
power  x",  n  being  a  positive  integer. 

The  nth  and  (^n  -\-  l)th  terms  of  such  a  power  series  are 

M„  =  a„_ja:"~i    and   m„^j  =  a„a:;". 

These  terms  may  be  positive  or  negative  even  if  a„_j  and  a„ 
are  both  positive,  since  x  may  be  positive  or  negative.  If 
we  wish  to  apply  the  ratio  test,  we  must  therefore  use  it  in 
the  extended  form  of  Art.  288.     We  have 


Wn+1     _ 

a„a;»      _ 

«n 

a„_ia;"-i 

«„-l 

Therefore,  the  series  will  be  convergent  or  divergent  for  a  given 
value  of  X,  according  as 

(1)  ■'■■    "• 


\x\  lim 

n — ^(x> 


is  less  than  or  greater  than  unity. 
If  we  write 


(2) 


lii 


■■n— 1 


1 

=  -     or 
r 


*n— 1 


lim 


^n-\ 


=  r. 


we  conclude  that  the  poiver  series 

«Q-|-  a^x  +  a^x^  -f  ••• 

is  convergent  for  those  values  of  x  for  which  \x\  <  r  and  diver- 
gent for  those  values  of  x  for  ivhich  |  a:  |  >  r. 


Art.  292] 


POWER  SERIES 


487 


It  may  happen  that 


Un- 


does not  approach  any  definite  limit,  finite  or  infinite.  In 
that  case  our  theorem  conveys  no  inforjnation.  But  if  such 
a  limit  exists,  we  have  the  following  results:  If  the  value 
of  r  obtained  from  (2)  is  zero,  the  power  series  is  divergent 
for  all  values  of  x  except  for  a:=0.  If  r  is  infinite,  the  series 
is  convergent  for  all  finite  values  of  x.  If  r  is  finite  and 
different  from  zero,  the  series  is  convergent  for  |a;|  <  r  and 
divergent  for  |  a;  |  >  r.  Our  test  tells  us  nothing,  however, 
as  to  whether  the  series  is  convergent  for  \x\  =  r^  that  is, 
when  x  =  ±r. 

Corollary.     If  a  'power  series  is  convergent  for  x  =  k,  it 
is  also  (ionvergent for  every  value  of  x for  ivliicli  \x\  <  |  A:  |. 

EXERCISE  CXXXVII 

Example  1.     Investigate  the  convergence  of  the  power  series 


4- 


+  J  +  T  + 


+  i-"  + 


Solution.     In  this  case  we  have 

1  1  a„ 


a-  = 


a„_,  = 


n  —  1     «„_i 


so  that 


lim 


J  =  l-i, 


=  1. 


Consequently  equation  (2),  Art.  292,  gives  r  =  1.     The  given  series  is  con- 
vergent for  all  values  of  x  for  which  |  x|  <  1  and  divergent  for  |  .i'  |  >  1. 

In  this  case  we  can  also  decide  what  happens  when  |  a:  [  =  1 .      For, 
if  X  =  +  1,  the  series  reduces  to 


UU1  + 

•   1      2      .} 


+  •  + 


This  is  the  harmonic  series  (see  Art.  282)  and  is  divergent.     If  j:  =  —  1, 
the  series  becomes 

-i  +  i-i  +  i-+  •••• 


488  INFINITE   SERIES  [Arts.  293,  204 

This  is  an  alternating  series  which  is  convergent  on  account  of  the 
theorem  of  Leibniz,  proved  in  Art.  290.  Thus  the  given  series  is  con- 
vergent when  I  x  I  <  1  and  also  when  x  —  —  1.  For  all  other  real  values 
of  X  the  series  is  divergent. 

Investigate  the  convergence  of  the  following  power  series. 

_     .        X    ,  X-    ,   x^  _        ,1  j:^  ,   1  ■  .3  x^  ,   1  •  .3  •  5  x'^  , 


1!      2  1      3!  •  2  3       2.4  r>      2.4-67 


7.  X  -  -  +  -  - 
3       5 


3!      5!      7!  3      3-^      38^ 

5.   •:?_£!+ ^^_£!+  ....  9.  1  +  x  +  2!x2  +  3!xH4!x4+  .... 

12        3       4 

293.  Equality  of  two-power  series.  The  following  theorem, 
which  is  really  an  extension  of  Theorem  F  of  Art.  126  has 
many  important  applications. 

Let  each  of  the  two-power  iSeries 

a(^+ a-^x  +  a^x^  +   ■■■  +  a„2:"  +  ••• 
and 

be  convergent  for  some  values  of  x  which  are  different  from  zero. 
If  the  sums  of  the  two  series  are  equal  to  each  other  for  all 
of  those  values  of  x  which  make  both  series  convergenU  the 
coefficients  of  like  powers  of  x  in  the  two  series  must  be  equal, 
that  is 

Aq  =  Jq,  aj  =  5^,  •  •  •,  a„  =  b^. 

A  rigorous  proof  of  this  theorem  would  require  a  rather 
long  chain  of  preliminary  theorems.  We  therefore  content 
ourselves  with  a  statement  of  the  theorem  witliout  giving 
a  proof. 

294.  Expansion  of  functions  as  power  series.  We  know 
from  the  theory  of  geometric  progressions  that  the  equation 

(1)  l  +  :,  +  :^:2+^+  ...  =      1_ 

1  —  X 


Art.  205]     EXPANSION    OF    RATIONAL    Fl'NCTIONS  489 

holds  for  all  values  of  x  for  which  1 2;  |  <  1.  We  may  ex- 
press this  by  saying  that  the  function 

(2)  .-^ 

\  —  X 

has  been  expanded  into  a  power  series,  namely, 

(3)  l4.:^+^:2_^^.3+  .... 

It  is  important  to  note  that  our  proof  of  equation  (1)  assures 
us  that  its  two  members  are  equal  to  each  other  for  all  of 
those  values  of  x  for  which  \x\  <  1,  but  not  for  any  other 
values  of  x.  In  fact  the  two  members  of  equation  (1)  are 
not  equal  to  each  other  for  2;  =  2.  F'or  tlie  left  member 
becomes  infinite  when  2;=  2,  while  the  right  member  is  equal 
to  —  1  when  a;  =  2. 

We  may  generalize  these  notions  as  follows.  When  a 
function  f{x)  is  given,  it  is  frequently  possible  to  find  a 
certain  power  series  which  is  equal  to  fix')  for  those  values 
of  x  for  which  the  power  series  is  convergent.  In  such 
cases  we  say  that  tlie  function  has  been  expanded  as  a  poiver 
series.  Tlie  equivalence  between  the  given  function  and  the 
power  series  can  never  hold  for  values  of  x  for  which  the  power 
series  does  not  converge. 

295.   Expansion  of   rational   functions.     We  may  use  the 

theorem  of  Art.  293  to  obtain  the  expansion  of  a  rational 
function,  whenever  such  an  expansion  exists,  by  the  method 
of  undetermined  coefficients.  (See  Art.  14:2.)  The  follow- 
ing examples  will  help  to  explain  this  method. 

1  —  X    • 
Example  1.     Expand  into  a  power  series. 

1  +  x'^ 

Sdlulinu.  It'  such  a  power  series  exists,  let  us  denote  its  coefficients 
(as  yet  unknown)  l»y  a^,  ui,  a.^,  and  so  on,  so  that 

1   —  X 

(1) ^  =  «o  +  ^1-^  +  "2-^"  +  "3-^^  +  ^^4^  +  •■•• 

If  this  equation  is  true  for  all  values  of  x  for  which  the  series  is  con- 
vergent, the  following  equation,  obtained  from  (1)  by  clearing  of  frac- 
tions, must  also  hold  for  all  such  values  of  x; 


490  INFINITE   SERIES  [Art.  295 

\  —  X  =  Gq  +  a^x  +  a^x'^  +  a^x'^  +  a^x*  +  ••• 
+  aQX-  +  a-^x^  +  rioX-*  +  •••• 
In  other  words,  we  must  have 

(2)  1  -  X  =  Qq  +  rtjx  +  («(,  +  rtj)^-  +  («!  +  f/3)x3+  (oj  +  a^)x*  +  •■• 

for  all  values  of  x  which  make  the  series  in  the  right  member  convergent. 
According  to  the  theorem  of  Art.  293,  this  can  be  so  only  if  the  coeffi- 
cients of  like  powers  in  the  two  members  of  (2)  are  equal  to  each  other. 
Consequently  we  conclude  that  (2)  can  be  true  only  if 

1  =  «o»   —  1  =  "d  0  =  «o  +  «2'  0  =  rtj  +  «3,  0  =  02  +  04,  •••, 
whence 

Oq  =  1,    «j  =  -  1,    flj  =  -  «0  =  -  1'    "3  =  -  "i  =  +  1,    O4  =  -   ^2  =  +   1»  — • 

Consequently  we  have 

(3)  i^^  =  1- x-x2  +  x3+  .r* , 

^  1  +  x^ 

if  there  exists  a  power  series  at  all  for  (1  —  .r)/l  +  x". 

In  this  particular  example  it  is  not  very  difficult  to  find  the  law  of  the 
coefficients,  to  prove  that  the  series  converges  for  !  .r|  <  1  and  to  prove 
that  equation  (3)  is  actually  true  for  all  such  values  of  x.  However, 
that  is  a  matter  with  which  we  are  not  primarily  concerned  just  now. 

Example  2.     Expand  '- into  a  power  series. 

x(x  —  1) 

Solution.     Let  us  try  to  use  the  same  method  as  in  Ex.  1  by  putting 

3 

(4)  — —  =  Qo  +  «i^  +  "2^^  +  •■■' 

x(x  —  1) 

whence,  clearing  of  fractions, 

3  =  —  UqX  —  «j.rj  —  aox"'  ••• 

+  (loX-  +  Oj.r^  +  •••. 

Equating  coefficients  of  like  powers  we  at  once  strike  a  contradiction, 
namely,  3  =  0.     Therefore  such  an  expansion  is  impossible  in  this  case. 

The  reason  for  this  impossibility  is  very  clear  from  (4).  The  function 
S/x(x  —  1)  has  X  =  0  as  a  pole  (see  Art.  139)  and  therefore  becomes 
infinite  when  x  approaches  zero  as  a  limit.  But  the  right  member  of  (4) 
remains  finite  for  x  =  0,  since  it  reduces  to  Qq.  Therefore  it  is  clear  by 
inspection  that  an  expansion  of  the  form  (4)  is  impossible. 

We  may,  however,  write 

3        ^3     1 
x(x  —  1)       X  X  —  1 


Art.  295]     EXPANSION   OF   RATIONAL    FUNCTIONS  491 

and  expand  the  second  fraction.     We  find 
1 


X-  1 
so  that 


—  \  —  X  —  X-  —  x^ 


^        =  _  ?  _  .3  -  .3  X  -  3  x2  -  3  x8  -  •• 


x(x-l)- 

This  is  not  an  ordinary  power  series  expansion  on  account  of  the  term 
—  3/x  which  occurs  in  it.  This  term  may  be  written  —  3  x"^.  It  in- 
volves a  negative  power  of  x  as  factor. 

These  examples  will  suffice  to  justify  the  following  gen- 
eral statement.  But  we  shall  not  attempt  to  give  a  formal 
proof  of  its  correctness. 

If  a  fractional  rational  function  of  x  does  not  have  x  =  0  as 
a  pole,  it  may  he  expanded  into  a  power  series  of  the  form 
a^  -f  a^x  +  a^x^  +  ••■ 

which  will  converge  for  some  non-vanishing  values  of  x,  but  not 
for  all  finite  values  of  x.  The  coefficients  of  this  series  may  be 
obtained  by  the  method  of  undetermined  coefficients. 

If  a;  =  0  is  a  pole  of  the  rational  function,  no  such  poioer 
series  exists,  but  the  function  may  be  expressed  as  the  sum  of 
such  a  power  series  and  certain  additional  terms,  each  of  these 
terms  having  a  negative  power  of  x  as  factor. 

The  following  remark  is  of  importance,  if  it  be  desired  to 
obtain  the  general  law  according  to  which  the  coef'licieuts  of 
the  expansion  are  formed,  so  as  to  be  able  to  judge  of  the 
convergence  of  the  resulting  series. 

In  order  to  be  able  to  recognize  the  general  laiv  of  the  co- 
efficients in  the  expansion  of  a  rational  function,  it  is  advisable 
to  express  the  function  as  a  sum  of  simple  partial  fractions 
(^see  Arts.  142-144)  ayid  then  to  expand  the  several  partial 
fractions  separately . 

EXERCISE  CXXXVIII 
Expand  the  following  rational  functions  in  powers  of  x.     Compute  at 
least  four  terms  of  the  expansion,  find  the  general  term  whenever  you 
can,  and  then  determine  the  values  of  x  for  which  the  resulting  series  are 
convergent : 


492  INFINITE  SERIES  [Art.  296 


1. 

1    -  X 

2. 

1 

I   +  X 

3. 

o 

1  ~x 

4. 

a 

\    -  X 

13. 

1 

1 

+ 

X 

6. 

J_ 

2 

— 

x 

7. 

J_ 

6" 

— 

X 

8. 

a 

h 

- 

X 

1 

9. 

h  +  X 

10. 

2  +  3a: 
1  -a;2 

11. 

1  +  a;  +  a;2 
1  -  X  +  a;2 

12 

2  a:  +  3  x2 

1  +  2  a:  +  3  x2 

2  + Sa- 
il +  xr 

16.  i-  -"^A 
(1  +  xY 

14.  ^ 15.  --! 

(1  +  a-)^         (1  +  x)3 

296.  Expansion  of  some  irrational  functions.  The  method 
of  Art.  295  may  be  used  with  very  small  modifications  in 
order  to  obtain  the  expansion  of  certain  irrational  functions. 


Illustrative  Example.     Expand  Vl  +  x  as  a  power  series. 
Solution.     Assume 


(1)  Vl  +  X  =  0(|  +  «ix  +  a.^x-  +  (i^x^  +  •••. 
Then  we  have,  squaring  both  members, 

(2)  1  +  X  =  rig-  +  2  itoU^x  +  2an«2^^  -^•  2  rt^agX^  +  ••• 

+  «i-X'^  +  2  fljOiX^  +  •••• 

Equating  coefficients  of  like  powers  of  x  in  the  two  members  of  (2),  we 
find 

(3)  1  =  a^'^,     1  =  2a^a^,     0  =  2a^a.-^  +  o,2,     0  =  2  a^n^  4  2a^a.^, 


and  so  on.  The  first  equation  (3)  gives  Aq  =±  1-  If  by  Vl  +  x  we 
mean  tlie  positive  square  root  of  1  +  x,  we  must  choose  rt^  =  +  1,  since 
the  positive  square  root  of  1  +  x  reduces  to  +  1  when  x  =  0.  Thus  we 
find  from  (3), 

"0=1-  «1   =   2'         ^^2  =  -  i.         ".■!  =  +   A. 

so  til  at 

vnr;-  =  1  +  I ,-  - 1  .r-  +  i,  .,-3  +  .... 

EXERCISE  CXXXIX 

Expand  the  following  functions  to  4  terms  : 

1.    Vl  +  x2.  4.    vTT^-.  7.    Vl  +  X  4-  X-'. 


2.    V;5  +  2x.  5.     v/1  -  X.  8.    xVl  + 


3.   V4^r^-.  6.   ^rT7^.  9.  ^^  +^. 

1-x 


Art.  297]  (IKNKRAL   BINOMIAL  EXPANSION 


493 


297.  The  expansion  of  (1  +  x)".  The  binomial  theorem 
(see  Art.  88 J  shows  us  that 

(1)  (]+.-)" 

whenever  n  is  a  positive  integer.  In  that  case  the  right 
member  of  (1)  contains  w  +  1  terms. 

If  n  is  a  negative  number  or  a  fraction,  we  may  still  form 
a  series  like  the  one  which  occurs  in  the  right  member  of 
(1),  but  in  all  such  cases  this  series  will  contain  an  infinite 
number  of  terms.  It  is  easy  to  show  that  this  series  will 
be  convergent  for  |a;j<  1  by  applying  the  ratio  test.  (Art. 
286.) 

We  have  in  fact 

""'  l-2.^...(k-l)k 


H-i 


_n(n-l)(n-2)  ...  (n-k+2') 


whence 


so  that 


1  .  2-3  ■••  (yfc-l) 
n  —k+1 


«i-i 


k  k 


lim 


=  1. 


'A-l  I 


Thus,  the  quantity  denoted  by  r  in  Art.  292  is  equal  to  1, 
and  we  conclude  from  the  principal  theorem  of  Art.  292  that 
the  series  is  convergent  for  all  values  of  x  for  which  |  a-j  <  1. 
Thus,  whenever  |a:|  <  1  the  right  member  of  (1)  is  a  con- 
vergent series  and  therefore  has  a  definite  meaning.  The 
left  member  of  (1)  also  has  a  definite  meaning.  (See  Arts. 
156  and  157.)  In  the  particular  case  when  w  is  a  i)Ositive 
integer  the  two  members  are  equal,  as  we  have  actually 
proved.  We  now  state  without  proof,  that  the  two  members 
of  (1)  are  equal  (for  \x\  <  1)  for  all  values  of  n.  This  gives 
us  the  general  binomial  theorem. 


494  INFINITE   SERIES  [Art.  297 

If  n  is  any  number^  not  necessarily  a  positive  integer^  we  may 
expand  (1  +  a;)"  according  to  the  formula 

(1)  (i  +  xy=\  +  ^^x+''"^'\-'^K'^+... 

n(n-l)(n-^^  ■■■{n-k+l)  ^ 
+  -  X  +  .... 

If  n  is  a  positive  integer^  this  expansion  consists  of  a  finite  num- 
ber of  terms  and  is  valid  for  all  values  of  x.  If  n  is  7iot  a  posi- 
tive integer^  the  expansion  will  be  an  infinite  series,  and  equation 

(1)  will  be  valid  for  all  values  of  x  which  are  numerically  less 
than  unity. 

In  Exercise  CXXXVIII,  Example  2,  Exercise  CXXXIX,  Example  4, 
and  the  illustrative  example  of  Art.  296,  we  found  a  few  terms  of  the  expan- 
sions for  -^  =  (1  +  x)-\  Vl  +  x  =  (1  +  x)i  v^rrr  =  (l  +  x)i     The 
1  +  X 

student  should  verify  that  the  expansions  obtained  by  him  for  these 
functions  are  in  agreement  with  the  results  which  would  be  obtained 
by  using  formula  (1).  This  may  be  regarded  as  a  partial  proof  of  (1). 
A  complete  proof  may  be  found  in  Dickson's  College  Algebra,  Chapter 
XV,  according  to  a  method  due  to  Euler.  The  most  convenient  proof, 
however,  depends  upon  methods  developed  in  the  calculus. 

Formula  (1)  may  also  be  used  to  compute  (a  +  by.     For 
we  may  write 

a-\-b  =  a(l-{-- 
\        ay 

(2)  Qa  +  by=a«(l+^y. 

The  second  factor  may  be  expanded  as  a  power  series  in 
X  =  b/a  by  means  of  (1)  if  1 5 1  <  |  a  |.  If  instead  1 6 1  >  |  a  |,  we 
write 


(a  +  by  =  b"(l  +  ^Y 


and  put  x=  a/b. 

This   method  is   very    convenient   for   computing    square 
roots,  cube  roots,  nth  roots  of  numbers. 


Art.  298]  EXPONENTIAL   SERIES  495 

Thus,  to  find  the  cube  root  of  30  we  write 

30  =  27  +  3  =  27(1  +  ^)=  27(1  +  i), 
so  that 

v^  =  </27y/TTl  =  3(1  +  i)i 
But  according  to  (1),  putting  n  =  J  and  x  =  ^,  we  find 

fi  +  i^^  =  i+i  1  ^m-i)  1  I  K^-i)a-2)  1 

V        9/  3  ■  y  1-2      81  1  •  2  •  3         729  ' 

the  various  terms  of  which  are  easily  computed. 

EXERCISE    CXL 

Expand  the  following  functions  to  five  terms  and  state  the  interval 
of  convergence  in  eacii  case  : 

1.    (1  +  3.0-1-  3.    vTT^.  5  1 


VI  -  3  X 

2.    (l  +  5a:)i  4.    (3  +  4  3:)-2.  6.    (2  -  x)^ 

Use  the  binomial  theorem  to  extract  the  following  roots  to  four  deci- 
mal places : 

7.  ViO.  9.    ^/1003.  11.    ^TdQ. 

8.  v^IjOI.  10.    -^US.  '        12.    </l3i. 

298.  Exponential  series.  The  exponential  function  e'  can 
be  expanded  as  a  power  series  of  the  following  form 

(1)  •"=1  +  ^  +  1^  +  17+  •••+^+-. 

as  may  be  proved  easily  by  the  methods  of  the  calculus. 
Although  we  shall  not  attempt  to  prove  this  formula,  we 
are  in  a  position  to  make  a  partial  check.  For,  if  we  put 
2:  =  1  in  this  equation,  we  find 

in  agreement  with  formula  (3)  of  Art.  182. 

The  student  will  find  it  easy  to  prove  that  the  exponential 
series  (1)  is  convergent  for  all  finite  values  of  x.  (See  Ex.  2, 
Exercise  CXXXVII.) 


x^      ofi 

X^        3^ 

=  2^  - 

2       3 

4       5 

496  INFINITE   SERIES  [Art.  299 

299.  Logarithmic  series.  It  can  be  shown  by  the  methods 
of  the  calculus  that 

(1)  log,(l+a-) 

..    +(_l)n-l^_+    ..., 

n 

where  \oge(l  +  x)  means  the  natural  logarithm  of  \  +  x. 
This  series  is  convergent  for  |2;|  <  1,  as  the  student  may 
verify  by  using  the  ratio  test.  (See  Ex.  5,  Exercise 
CXXXVII.) 

Theoretically  (1)  may  be  used  to  calculate  the  natural  logarithm  of 
a  number  1  +  x  whenever  |  x  |  <  1.  But  actually  the  difficulty  arises  that 
the  series  (1)  converges  very  slowly,  so  that  a  very  large  number  of  terms 
would  be  required  in  order  to  obtain  a  fairly  accurate  result. 

It  is  easy,  however,  to  find  a  more  convenient  formula  for  the  calcula- 
tion of  natural  logarithms.     From  (1)  we  obtain 

(2)  log,(l-x)  =  -x-f-^-5^-  ....  lxl<l. 

J        o        4 

Combining  (1)  and  (2),  by  subtraction  we  find 

(3)  log.  (1  +  x)  -  log,  (1  -  x)  =  2 (x  +  I'  +  I  +  •  •  ■) , 

a  formula  which  is  valid  for  |  x|  <1.     The  left  member  of  (.3)  is  equal 

to  the  logarithm  of  ^-^t^"  (Theorem  VIII,  Art.  106),  so  that 
1  —  X 

(4)  ,„g,^=o(.  +  |^^^...). 

If  X  is  positive  and  less  than  unity,  (1  +  x)/(l  —  x)  is  a  positive  num- 
ber greater  than  1,  and  we  may  write 

1  +  a:  _  m  +  1 
1  —  a:  m 

which  gives 

1 


2»«  +  1 
Thus  (4)  becomes 

?n  +  l_ori         ,  1  I 1 L 


(6)        log/i^±i=.2r.^ 
m  \-2  m  +  1 


3{2m+iy      5(2  m+ 1)5 


This  formula  is  very  convenient  for  calculation.     If  we  put  m  =  1,  we  find 
log,  2  =  0.0931+;  from  the  result  obtained  for  ?/4  =  2  we  find  log^  3  =  1.0980+. 


Art.  299]  LOGARITHMIC    SERIES  497 

After  we  have  computed  the  natural  logarithms  in  this  way  we  may 
compute  the  common  logarithms  by  the  method  of  Art.  179.     We  have 

logiox=    "^'^  • 
log.  10 

Thus,  the  common  logaritlini  of  x  is  obtained  by  multiplying  the  natural 
logarithm  of  x  by  the  so-called  modulus 

1 


M  = 


lOgr.  10 


The  value  of  .1/  may  be  obtained  by  putting  m  =  9  in  equation  (6)  and 
making  use  of  the  value  log,  3  which  has  already  been  obtained.  We 
should  find  in  this  way,  to  five  significant  figures, 


M  =  0.43429. 


EXERCISE    CXLI 


1.   Compute  the  natural  and  common  logarithms  of  the  numbers  from 
1  to  10  to  four  decimal  places. 


APPENDIX 

TABLE   1.     FOUR   PLACE   LOGARITHMS    OF   NUMBERS 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

03:34 

0374 

11 

0414 

0453 

0492 

0531 

05(59 

0607 

0645 

0682 

0719 

0755 

12 

0792 

0828 

0864 

0899 

09:i4 

0969 

1004 

1038 

1072 

1106 

13 

1139 

1173 

1206 

12:59 

1271 

130:5 

1335 

13(57 

1399 

1430 

14 

1461 

1492 

1523 

1553 

1584 

1614 

1644 

1673 

1703 

1732 

15 

1761 

1790 

1818 

1847 

1875 

1<X)3 

1931 

1959 

1987 

2014 

16 

•2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2279 

17 

2:504 

2330 

2355 

2:580 

2405 

2430 

2455 

2480 

2504 

2529 

18 

2553 

2577 

2601 

2625 

2648 

2672 

2(595 

2718 

2742 

2765 

19 

2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

21 

3222 

3243 

32()3 

3284 

3304 

3324 

3345 

3365 

3385 

3404 

22 

3424 

3444 

3464 

3483 

3502 

3522 

3541 

3560 

3579 

3598 

23 

3617 

3636 

3655 

3674 

3692 

3711 

3729 

3747 

3766 

3784 

24 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

26 

4150 

4166 

4183 

4200 

4216 

42:52 

4249 

42(55 

4281 

4298 

27 

4314 

4330 

4346 

4:562 

4378 

4393 

4409 

4425 

4440 

445(5 

28 

4472 

4487 

4502 

4518 

4533 

4548 

45(54 

4579 

4594 

4(509 

29 

4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

31 

4914 

4928 

4942 

4955 

49(>9 

4983 

4997 

5011 

5024 

5038 

32 

5051 

5065 

5079 

5092 

5105 

5119 

51:32 

5145 

5159 

5172 

33 

5185 

5198 

5211 

5224 

52:57 

5250 

5263 

5276 

5289 

5302 

34 

5315 

5328 

5340 

5:353 

5366 

5378 

5391 

5403 

5416 

5428 

35 

5441 

5453 

5465 

5478 

5490 

5502 

5514 

5527 

5539 

5551 

36 

55()3 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5658 

5670 

37 

5682 

5694 

5705 

5717 

5729 

5740 

5752 

57(53 

5775 

5786 

38 

5798 

5809 

5821 

5832 

5843 

5855 

5866 

5877 

5888 

5899 

39 

5911 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010 

40 

6021 

6031 

6042 

6053 

(50(54 

(i075 

6085 

(509(5 

6107 

6117 

41 

6128 

(>13« 

6149 

61(50 

(5170 

(5180 

6191 

(5201 

(5212 

6222 

42 

()232 

6243 

(J253 

(52(53 

6274 

(5284 

6294 

(i:504 

(5314 

6325 

43 

6335 

6345 

6355 

6:565 

(5375 

(i:585 

6395 

(5405 

6415 

6425 

44 

6435 

6444 

6454 

6464 

6474 

(5484 

6493 

(5503 

6513 

6522 

45 

6532 

()542 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

46 

(5628 

6637 

6(i46 

6656 

(!()()5 

(5(575 

(5(584 

6(593 

6702 

6712 

47 

(1721 

6730 

6739 

6749 

6758 

(5767 

6776 

(5785 

(5794 

6803 

48 

()812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

(5884 

6893 

49 

6902 

()911 

(5920 

6928 

69:57 

(;94() 

6955 

6<X)4 

(5972 

6981 

50 

6990 

6998 

7007 

7016 

7024 

70:5:5 

7042 

7050 

7059 

7067 

51 

7076 

7084 

7093 

7101 

7110 

7118 

7126 

7i;35 

7143 

7152 

52 

7160 

7168 

7177 

7185 

7193 

7202 

7210 

7218 

722(5 

7235 

53 

7243 

7251 

7259 

72(57 

7275 

7284 

7292 

7300 

7:308 

7316 

54 

7324 

7332 

7:540 

7:548 

7356 

7:564 

7372 

7:380 

7388 

7396 

498 


APPENDIX 


499 


N 

0 

1 

2 

3 

4 

5 

6 

' 

8 

9 

55 

7404 

7412 

7419 

7427 

74:!5 

744:5 

7451 

7459 

7466 

7474 

56 

74.S2 

7490 

7497 

7505 

7513 

7520 

7528 

75:5(5 

7543 

7551 

57 

7559 

756(j 

7574 

7582 

7589 

7597 

7604 

7(512 

7619 

7627 

58 

ny.n 

7()42 

7(549 

76."i7 

7()()4 

7(572 

7679 

7(58(5 

7()94 

7701 

59 

770<) 

7716 

7723 

7731 

7738 

7745 

7752 

77(50 

77(57 

7774 

60 

7782 

7789 

77% 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

61 

785:? 

7860 

78(i8 

7875 

7882 

7889 

78!K) 

7903 

7910 

7917 

62 

7!I24 

7931 

7938 

79-15 

79o2 

7959 

7i)(i6 

797:5 

7980 

7987 

63 

7!l*>;5 

8000 

8007 

8014 

8021 

8028 

80:55 

8041 

8048 

8055 

64 

80ti2 

8069 

8075 

8082 

8089 

8096 

8102 

8109 

8116 

8122 

65 

81 2<) 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

66 

.Sl<t5 

8202 

8209 

8215 

8222 

8228 

82:?5 

8241 

8248 

8254 

67 

82(11 

8267 

8274 

8280 

8287 

8293 

82<t9 

8:506 

8312 

8319 

68 

8;i25 

8331 

8338 

8344 

8:551 

8357 

83(53 

8:570 

8:576 

8382 

69 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

70 

8451 

8457 

84(53 

8470 

8476 

8482 

8488 

84f>4 

8500 

8506 

71 

85i;{ 

8519 

8525 

8531 

85:57 

8543 

8549 

8555 

8561 

8567 

72 

8573 

8579 

8585 

8591 

8597 

8()03 

8609 

8(515 

8(521 

8627 

73 

8(>:5;i 

8639 

8645 

8(551 

8(>57 

8(5(i3 

86(59 

8675 

8(i81 

8686 

74 

8(i'J2 

8698 

8704 

8710 

8716 

8722 

8727 

87:53 

8739 

8745 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

76 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

77 

88«5 

8871 

8876 

8882 

8887 

889;$ 

8899 

8<K)4 

8910 

8915 

78 

8f)21 

8927 

8932 

89:« 

8943 

8;>49 

8954 

85K50 

8965 

8971 

79 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9025 

80 

9031 

903<) 

9042 

9047 

905:5 

9058 

9063 

i)0(i9 

9074 

9079 

81 

1KXS5 

9090 

909(5 

9101 

9106 

9112 

9117 

9122 

9128 

91 ;« 

82 

9138 

9143 

9149 

9154 

9159 

9165 

9170 

9175 

9180 

9186 

83 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

t)238 

84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

85 

9294 

92<)9 

9:504 

9:509 

9:515 

9320 

9325 

9:i:50 

9335 

9340 

86 

9;M5 

9350 

9:555 

9:5()0 

9:5(55 

9370 

9375 

9:580 

9:585 

9390 

87 

9395 

9400 

i)405 

<t410 

!H15 

(M20 

f>425 

94:50 

m:55 

9440 

88 

!H45 

9450 

9455 

<H()0 

94(55 

m(i9 

<)474 

9479 

!>484 

9489 

89 

9491 

9499 

9504 

9.509 

9513 

9518 

9523 

9528 

953;5 

9538 

90 

9542 

9547 

9552 

9557 

9562 

95(!6 

9571 

9576 

9581 

9586 

91 

95!  K) 

9595 

9(500 

9605 

<H)09 

!h;u 

i)619 

9()24 

<)(528 

9(533 

92 

9<)38 

9()43 

!«547 

fH552 

9(;57 

!Hi(il 

9(5(5(5 

;H571 

i)675 

9680 

93 

9(585 

9689 

9(594 

9(59;t 

97015 

9708 

9713 

9717 

9722 

9727 

94 

9731 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

9773 

95 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

96 

9823 

9827 

9832 

98:5(5 

9841 

9845 

9850 

9854 

9859 

98(53 

97 

98()S 

9872 

9877 

9881 

988(1 

98  W 

98ft4 

9899 

<)903 

9908 

98 

9912 

9917 

9921 

992(; 

99:50 

99:54 

<«»:59 

9!»4:5 

9^8 

<»952 

99 

995() 

9iKil 

99(55 

99(59 

9974 

997.S 

9983 

9987 

9991 

9996 

500 


APPENDIX 


TABLE   2.     AMERICAN  EXPERIENCE   TABLE   OF 
MORTALITY 


Number 

Ni-M- 

BEl". 

Yearly 
Probabil- 

Yearly 
Probabil- 

us 
o 

Number 

Num- 
ber 

Yearly 
Probabil- 

Yearly 
Probabil- 

< 

Living 

Dying 

ity  OF 
Dying 

ity  OP 
Living 

■< 

Living 

Dying 

ity  OF 
Dying 

ity  OF 
Living 

10 

100  000 

749 

.007  490 

.992  510 

53 

66  797 

1091 

.016  333 

.983  667 

11 

99  251 

746 

.007  516 

.992  484 

54 

65  706 

1  143 

.017  396 

.982  604 

12 

98  505 

743 

.007  543 

.992  457 

55 

64  5(i3 

1  199 

.018  .571 

.981  429 

13 

97  7()'2 

740 

.007  5()9 

.992  421 

56 

63  364 

1  260 

.019  885 

.980  115 

14 

97  022 

737 

.007  596 

.992  404 

57 

62  104 

1325 

.021  335 

.978  665 

15 

96  285 

735 

.007  634 

.992  .366 

58 

60  779 

1  394 

.022  936 

.977  064 

16 

95  550 

732 

.007  661 

.992  .339 

59 

59  385 

1468 

.024  720 

.975  280 

17 

9-1  818 

729 

.007  688 

.992  312 

60 

.57  917 

1546 

.026  693 

.973  307 

18 

91  089 

727 

.007  727 

.992  273 

61 

56  371 

1  628 

.028  880 

.971  120 

19 

93  3(J2 

725 

.007  765 

.992  235 

62 

54  743 

1713 

.031  292 

.968  708 

20 

92  637 

723 

.007  805 

.992  195 

63 

53  030 

1  ,'  CO 

.033  943 

.9()6  057 

21 

91  914 

722 

.01(7  855 

.992  145 

64 

51  230 

1  8.-9 

.0.36  873 

.963  127 

22 

91  192 

721 

.007  906 

.992  094 

65 

49  341 

1  980 

.040  129 

.959  871 

23 

90  471 

720 

.007  958 

.992  042 

66 

47  361 

2  070 

.043  707 

.956  293 

24 

89  751 

719 

.008  Oil 

.991  989 

67 

45  291 

2  158 

.047  647 

.952  353 

25 

89  032 

718 

.008  065 

.991  935 

68 

43  133 

2  243 

.052  002 

.947  998 

26 

88  314 

718 

.008  130 

.991  «70 

69 

40  890 

2  321 

.056  762 

.943  238 

27 

87  596 

718 

.008  197 

.991  803 

70 

38  5()9 

2  391 

.061  993 

.938  007 

28 

8»)  878 

718 

.008  264 

.991  73t) 

71 

36  178 

2  448 

.067  665 

.932  335 

29 

86  160 

719 

.008  345 

.991  655 

72 

33  730 

2  487 

.073  733 

.926  267 

30 

85  441 

720 

.008  427 

.991  573 

73 

31  243 

2  505 

.080  178 

.919  822 

31 

84  721 

721 

.008  510 

.991  490 

74 

28  738 

2  501 

.087  028 

.912  972 

32 

84  000 

723 

.008  607 

.991  393 

175 

26  237 

2  476 

.094  371 

.905  629 

33 

83  277 

726 

.008  718 

.991  282 

76 

23  761 

2  431 

.102  311 

.897  689 

34 

82  551 

729 

.008  831 

.991  169 

77 

21330 

2  369 

.111  064 

.888  936 

35 

81  822 

732 

.008  946 

.991  054 

78 

18  961 

2  291 

.120  827 

.879  173 

36 

81  090 

737 

•009  089 

.990  911 

79 

K!  670 

2  196 

.131  734 

.868  2(i6 

37 

80  353 

742 

.009  234 

.990  7()6 

80 

14  474 

2  091 

.144  4()() 

.8.55  534 

38 

79  (ill 

749 

.009  408 

.990  592 

81 

12. 38.3 

1  964 

.158  605 

.841  395 

39 

78  8(i2 

756 

.009  586 

.990  414 

82 

10  419 

1816 

.174  297 

.825  703 

40 

78  10() 

765 

.009  794 

.990  20() 

83 

8  603 

1  648 

.191  561 

.808  439 

41 

77  341 

774 

.010  008 

.9S9  992 

84 

()  955 

1  470 

.211  359 

.788  641 

42 

76  5()7 

785 

.010  2.52 

.989  748 

85 

5  485 

1  292 

.235  552 

.7(54  448 

43 

75  782 

797 

.010  517 

.989  483 

86 

4  193 

1  114 

.265  681 

.734  319 

44 

74  985 

812 

.010  829 

.989  171 

|87 

3  079 

933 

.303  020 

.()96  980 

45 

74  173 

828 

.011  163 

.988  837 

88 

2  146 

744 

.346  692 

.(353  308 

46 

73  345 

848 

.011  562 

.988  4.38 

89 

1402 

555 

.395  863 

.(i04  137 

47 

72  497 

870 

.012  000 

.988  000 

90 

847 

385 

.454  545 

.545  455 

48 

71  627 

896 

.012  509 

.987  491 

91 

462 

246 

.532  466 

.467  5:M 

49 

70  731 

927 

.013  106 

.986  894 

92 

216 

137 

.634  259 

.365  741 

50 

69  804 

962 

.013  781 

.98()  219 

93 

79 

58 

.734  177 

.265  823 

51 

(i8  842 

1001 

.014  541 

.985  4.59 

94 

21 

18 

.8,57  143 

.142  857 

52 

67  841 

1044 

.015  389 

.984  611 

95 

.3 

3 

1.000  000 

.000  000 

INDEX 


References  are  to  }-)ages 


Abel 207,  2(54 

Abscissa 70 

Absolutely  convergent  series  483 
Absolute  value,   of  a  complex 

number 39 

of  a  directed  line-segment .     22,  32 

Acceleration 04 

Addition,  of  complex  numbers  .  38 

of  positive  integers   ....  1 

of  positive  and  negative  iniin- 

bers 24 

of  rational  numbers  ....  11 

of  vectors 36 

Ahmes 52 

Algebraic,  function 203 

.solution  of  equations 

187,  107,  204,  207 

Algorithm.  Euclid's      ....  0 

Alternating  series 484 

Amount 306 

Amplitude,  of  a  complex  number  36 

of  a  directed  line-segment .     .  32 

Annuity 308 

Approximation, 

Newton's  method  of  .     .     .  157 

Archimedes 182 

principle  of 182 

Area 61 

Argand 53 

Argument,  of  a  complex  number 

32,  36 

of  a  directed  line-segment  32 

Aristotle 281 

Arithmetic,  means 84 

progressions 81 

Associative  law,  of  addition  2 

of  multiplication 2 


Assumptions 3 

Asymptote 232 

Atmospheric  density  and   pres- 
sure       313 


Ball 

Beman 

Between  .... 
Bhaskara  .... 
Binomial  theorem  . 
Bombelli  .... 
Bounded  variable 

Boyle 

Briggs 

Briggsian  logarithms 
Biirgi 


88, 


282 

186 

11 

53 

493 

53 

448 

246 

282 

282 

303 


Cantor 19 

Cardano 63, 

Cardan's  formulas 

Case 

Cauchy 208, 

Changing  sign  of  roots     .     .     . 

Characteristic 

Cof  actor 337, 

Combination 

Common  difference 

Common  ratio 

Commutative  law,  of  addition  . 

of  multiplication 

Comparison  tests 

Complex  numbers 

addition  of 

conjugate     

division  of 

equality  of 

multiplication  of 


,  53 

201 

201 

367 

264 

154 

282 

384 

362 

81 

86 

1 

2 

471 

34 

38 

61 

47 

36 

44 


501 


502 


INDEX 


Complex  numbers  —  Continued 

polar  form  of 44 

subtraction  of 40 

Complex  roots  of  unity     .     .     .  189 
Components,  of  a  directed  line 

segment 33 

of  a  displacement      ....  37 

Composite,  number      ....  3 

quadratic  function    ....  392 

Compound  events 368 

Compound  interest  ...       306,  309 

Compound  interest  law    .     .     .  311 

Conditional  equation    ....  211 

Conditionally  convergent  series  484 

Conies 407 

Conjugate  complex  numbers      .  51 

Conjugate  of  a  quadratic  surd   .  114 

Constant 55 

Construction     of    regular  poly- 
gons      193 

Continuity,  of  a  function  .     .  166,  457 

of  an  integral  rational  func- 
tion       156,  458 

of  a  fractional  function      .     .  458 

Convergent  series 466 

Cooling  bodies 315 

Coordinates 70 

Cubic  equation 197 

reduced  form  of 197 

Cyclotomic  equations  ....  193 

D'Alembert 209 

Dalton 347 

Dampened  vibrations  ....  312 

Dedekind 19,  53 

Defective 244 

Degree  of  integral  rational  func- 
tion             129,  391 

Delian  problem 186 

De  Moivres  formula    ....  189 

Denominator 8 

Dense 17 

Density 62 

variation  of  density  in  atmos- 
phere    313 

Dependent  events 368 

Dependent  variable      ....  55 


Depressed  equation      ....  166 
Derivative,  of  a  function  .     .     .  139 
of  an  integral  rational   func- 
tion       146 

of  higher  order 146 

Descartes 53,  168 

Descartes's  rule 168 

Determinant,  of  second  oi'der   .  325 

of  third  order 334 

of  ni\\  order 377 

Dickson      180,  202 

Difference 1,  2 

Dimension 61,  391 

Dimensional  symbols  ....  QQ 
Diminishing  roots  of  an  equa- 
tion        149 

Directed  line,  positive  sense  of  .  21 
Directed     line-segment     of     a 

directed  line 21 

absolute  value  of 22 

measure  of 22 

numerical  value  of    ...     .  22 

origin  of 21 

standard  position  of  ...     .  22 

terminus  of 21 

Directed  line-segment  of  a  plane  31 

absolute  value  of 32 

amplitude  of 32 

argument  of 32 

components  of 33 

modulus  of 32 

origin  of 32 

standard  position  of  ...     .  32 
terminus  of      ...     .           .32 
Discriminant,       of       quadratic 

equation 106 

of  quadratic  function  of  two 

variables 395 

Displacement 36 

components  of 37 

resultant  of  two 37 

Distributive  law 2 

Divergent  series 467 

Dividend 4 

Division,  by  a  positive  or  nega- 
tive divisor 28 

by  zero 29 


INDEX 


503 


Division,  of  complex  numbers  .  47 

of  integers 4,  n 

of  positive  rational  numbers  .  14 

synthetic 13.j 

uniqueness  of 15 

Divisor 4 

greatest  common 5 

of  an  integer ;> 

Domain  of  convergence     .     .     .  48() 

Dopplers  principle      ....  80 

Duplication  of  the  cube    .     .     .  18(5 

e 208 

numerical  value  of  e      .     .     .  299 

Eliminant 380 

Ellipse 404 

Equation,  conditional  ....  211 

of  a  straight  line 75 

linear,    quadratic,    etc.      See 
adjectives. 
Equivalence,  of  fractional  equa- 
tions      243 

of  in-ational  equations  .     .     .  262 

of  quadratic  equations  .     .     .  00 
of    systems    of    simultaneous 

equations 411 

Errors  of  observation   ....  375 

Euclid 6 

Euclid's  algorithm       ....  6 
Euler     ....     207,  209,  435,  494 

Events 366 

compound 368 

dependent 368 

exclusive     • 3(i8 

independent 3()8 

Exclusive  events 368 

Expansion,  binomial    ....  493 

Taylor's 148 

Expectation 374 

Exponential,  equations     .     .     .  301 

functions 274 

series 405 

Exponents,  fractional,  negative. 

and  vanishing 265 

irrational 270 


Factor,  highest  common    . 


229 


of  an  integer 3 

of  proportionality      ....  67 

prime 6 

Factored  form  of  rational  func- 
tion       2;S3 

Factorial 353 

Factor  theorem 132 

Falling  bodies 66 

False  position,  mt'thod  of     .     .  158 

Fermat 435 

Ferrari 207 

Ferro 201 

Field 114 

Floating  spheres 182 

Follows 11 

Force 121 

dimension  of 124 

Foster '  .     .     .     .  200 

Fourth  order,  equation  of           .  204 

Fraction 8 

rational 226 

reduction  to  lowest  terms  .     .  10 

Fractional  equations    ....  241 

Fractional  exponents  ....  267 

Fractional  rational  functions     .  226 

improper 226 

in  lowest  terms 228 

proper 226 

reduction  to  lowest  terms  .     .  229 

Function 55 

linear,    quadratic,    etc.     See 

adjectives. 

Functional  notation      ....  131 

Fundamental  laws  .     .1.2.  .],  12,  13 

for  coiiiplex  imiiibers     ...  51 

Fundamental  theorem  of  algebra  208 

Galois 208 

Gauss 5:],  104,  209 

Gauss  plane 63 

Gay-Lussac 246 

General  term  of  a  sequence  .     .  429 
Geometric  means     .     .     .     .     87,  88 

Geometric  progression       ...  86 

common  ratio  <if 86 

.sum  of 87 

with  infinitely  many  terms     .  89 


504 


INDEX 


Girard 203,  209 

Greater 11 

Greatest  common  divisor      .     .        5 

Gunter 290 

Gunter's  scale 289 

Harmonic  means 85 

Harmonic  progression  ....  85 

Hermite 208 

Higher  progressions      ....  431 

Highest  common  factor     .     .     .  229 

Homogeneous  linear  equations, 

with  two  unknowns  ....  326 

with  three  unknowns    .     .     .  345 

with  n  unknowns      ....  387 

Horner 164 

Horner's  method 164 

Hyperbola 232,  405 

asymptotes  of  rectangular      .  232 

i 35,  50 

Identity 211 

Imaginary  numbers  ....  50 
Independent  events  ....  368 
Independent  variable  ....  55 
Indeterminate  forms    ....     460 

Index  laws 265 

Indirect  analysis 348 

Induction,  mathematical  .  134,  434 
Infinite  roots  of  an  equation      .     224 

Infinite  series 466 

convergent 466 

divergent 467 

oscillating •     467 

Infinitesimal 448 

Infinity 447 

Inflection,  point  of 218 

Integer 1 

Integral  rational  function,  of  one 

variable 129 

of  two  variables 391 

Interest,  simple 306 

compound ,306,  309 

Interpolation 285,  431 

Inversion 359 

Irrational,  equations    ....     261 
exponents 270 


functions 249 

numbers 17 

roots  of  quadratic  equation    .  112 

Jacobi 264 

Kelvin 312 

Klein 186 

Lagrange   208,  299 

Legendre 435 

Leibnitz     . 139 

Lemma 133 

Length       61 

Leonardo  of  Pisa 53 

Less 11 

Light,  transmission  of      .     .     .  314 

Limit 444 

Linear  equations,  with  one  un- 
known        78 

with  two  unknowns  ....  321 

with  three  unknowns     .     .     .  342 

with  n  unknowns      ....  387 

Linear  function,  of  one  variable  77 

of  two  variables 320 

Logarithm 277 

common 281 

natural 296,  299 

Logarithmic,  decrement  .     .     .  312 

increment 312 

paper .  317 

scale 289 

series 496 

tables 498 

Lowest  terms 10 

Mannheim 292 

Mantissa 282 

Mariotte 246 

Mass 62 

Mathematical  Induction  .       134,  434 
Maximum  or  Minimum,  of  an 

integral  rational  function  .  173 
of  a  quadratic  function  ...  97 
Measure  of  a  directed  line-seg- 
ment      22 


INDEX 


505 


Method,  Horner's 164 

Newton's 157 

of  false  position 158 

of  mathematical  induction  134,  434 

of  small  corrections  ....  420 

of  UQcletermined  coefficients  .  236 
Minimum.     See  .Maximum. 
Minor  of  a  determinant, 

of  the  third  order      ....  335 

of  the  Jith  order 382 

Minuend     2 

Modulus 32,  39 

Monotonic  laws 2 

Mortality,  table  of  .     .     .       373,  500 

Motion,  uniform 120 

of  a  projectile 126 

Multiple,  lowest  common      .     .  6 
poles  of  a  rational  function    .  231 
roots  of  an  equation ....  175 
zeros  of  a  rational  function    .  230 
Multiplication,  of  complex  num- 
bers       44 

of  positive  integers    ....  1 
of  positive  and  negative  num- 
bers       26 

of  rational  numbers  ....  13 
of   the  roots  of   an   equation 

by  ?» 153 

Multiplicity  of  a  zero  or  pole 

230,  231 

Napier 303 

Napierian  base 298 

Natural  logarithms      .     .      296, 298 

Negative,  exponents    ....  268 

numbers 20 

Newton 139,  141 

Newton's  laws  of  motion        121,  123 

Newton's  method 157 

Normal    form    for    expressions 

which     involve     (luadratic 

surds 260 

Normalization   of  a  system  of 

simultaneous  quadratics      .  413 
Numbers,     negative,     complex, 

etc.     See  adjectives. 

Numerator 8 


Numerical  value 22 

NuSez 59 

Order,  notion  of 80,  352 

of  an  integral  rational  func- 
tion of  one  variable    .     .     .  129 
of   an  integral  rational  fimc- 
tion  of  two  variables  .     .     .  391 

Ordinate 70 

Origin,  of  coordinates  ....  70 

of  line-segment 11 

of  directed  line-segment  on  a 

directed  line 21 

of  directed  line-segment  in  a 

plane 32 

of  scale  of  integers    ....  7 

Oughtred 290 

Parabola 94 

vertex  and  axis  of     ...     .  98 

Partial  fractions 234 

Periodic  decimal 91 

Permanence,  principle  of      .     .  268 

Permutation 362 

even  or  odd 359 

principal 359 

Plane,  of  the  complex  variable  .  53 

Gauss 53 

Point  of  inflection 218 

Polar  form  of  a  complex  number  44 
Pole  of  a  rational  function    .     .  231 
Polygons,  construction  of  regu- 
lar       .     .  193 

Pound  of  force 125 

Poundal 125 

Power  function 272 

Power  series 486 

Precedes 11 

Premium 373 

Pressure,  in  the  atmosphere  .  .  .  313 

of  gases 245 

Prime,  factors 6 

number  .     .     .- 3 

relatively 6 

Principal 306 

Principal  value,  of  Vx     .     .     .  250 

of  y/x^ 263 


506 


INDEX 


Principle  of  permanence  .     .     .  268 

Probability 366 

Product 1,  2 

geometric  construction  of  .     .  13 

Progressions,  arithmetic  ...  81 

geometric 86 

harmonic 85 

Ptolemy 52 

Pythagoras 19,  53 

Pythagoreans 19 

Quadratic  equations     ....  99 

complex  roots  of 104 

discriminant  of 100 

equivalence  of       ....   99,  100 

pure 109 

rational  and  irrational  roots  of  111 

roots  of 101,  103 

with  zero  roots 109 

Quadratic  function  of  one  vari- 
able       94 

graph  of 95 

maximum  or  minimum  of       .  97 

standard  form  of 94 

zeros  of 99 

Quadratic  functions  of  two  vari- 
ables    392 

composite  or  non-composite   .  392 

discriminant  of 395 

Quadratic  surds 112 

Quartic  equation 204 

Radicals,  properties  of     .     .     .  254 

Rate  of  interest 306 

Rational  function 226 

Rational  numbers 8 

Rational  roots, 

of  quadratic  equation    .     .     .  Ill 

of  equations  of  higher  degree  179 
Rationalizing   the  denominator 

114,  260 

Ratio  test       474 

Real  numbers 50 

Reciprocal 15 

Redundant 244 

Reflection        252 

Regula  falsi 158 


Regular  polygons,  construction 

of 193 

Relatively  prime 6 

Remainder 4 

Remainder  theorem      ....  134 

Resultant,  of  two  displacements  37 

of  two  equations 389 

Riemann 264 

Rolle's  theorem 175 

Roots  of  quadratic  equation  101,  103 

Scale,  notion  of 292 

of  positive  and  negative  num- 
bers      20 

Second  derivative 148 

Semi-logarithmic  paper    .     .     .     316 

Sequence •     .   80,  429 

Series 433 

Simultaneous  equations 

322,  407,  409 

Skinner 374 

Slide  rule 290 

Slope 73 

of  tangent  of  a  curve     .     .     .     136 

Smith 186,  303 

Solution  of  an  equation    .       321,  399 

Specific  gravity 63 

Spheres,  floating 182 

Standard  logarithmic  curve  .     ,     295 
Standard  position  of  a  line-seg- 
ment     22,  32 

Subtraction,   of  complex  num- 
bers      40 

of  positive  integers   ....         1 
of  rational  numbers  .     ...       11 

of  vectors 40 

Subtrahend •        2 

Sum 1,  12 

geometric 37 

Summation  sign 437 

Surds,  quadratic 112 

Symmetric  functions   ....     221 
Synthetic  division 135 

Table  of  logarithms     ....    498 

use  of 285 

Table  of  mortality  .     .     .      373,  500 


INDEX 


501 


Tartaglia 201 

Taylors  expansion       ....  148 
Terminus  of  line-segment     12,  21,  32 

Time (>2 

Transcendental  function  .     .     .  2(58 

Transmission  of  light  ....  314 

Trial 3Gt> 

Trisection  of  an  angle       .     .     .  186 

Uniform  motion   on   a  straight 

line 120 

Uniformly  accelerated  motion  .  05 

Uniqueness  of  division     ...  15 

Van  der  Waals 240 

Vanishing  roots 222 

Variable 55 

bounded      .     • 448 

independent  or  dependent      .  55 

which  remains  finite      .     .     .  448 

Variation 50 

constant  of 57 

Variations  of  an  equation     .     .  109 


Vector 39 

Vector  addition 30 

Velocity 03 

Vernier 5!» 

Vibrations,  dampened       .     .     .  313 

Vieta 2<)3 

Volume 01 

Weierstrass 19,  53,  204 

Weld 375 

Wessel 53 


x-axis 

jr-intercept  of  a  line 


y-axis 

1/- intercept  of  a  line 
Young    


70 
70 

70 

75 

180 


Zero 20 

as  exponent 267 

division  by 29 

of  a  function 78 


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